In [1]:
# python standard
import time
# third-party
import pandas as pd
import numpy as np
# locals
from core.constantes import *
from core.poco_quadrado_finito import PocoQuantico
Configurações gráficas¶
In [2]:
import locale
locale.setlocale(locale.LC_NUMERIC, "pt_BR.UTF-8")
import matplotlib.style
import matplotlib as mpl
mpl.style.use('classic')
%matplotlib inline
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
plt.style.use('mestrado')
Inicia um poço quântico¶
In [3]:
poco_100 = PocoQuantico(well_length=100.0, well_height=1.0, N=2048, dt=1e-18)
Este é o potencial que vamos usar:
In [4]:
fix, ax = plt.subplots(1,1, figsize=(18,6))
ax.plot(poco_100.z_ang, poco_100.v_ev)
ax.set_xlabel(r'z (A)')
ax.set_ylabel(r'V (eV)')
plt.show()
que foi iniciado com as seguintes propriedades:
- well_length=100.0, largura de 100.0 Angstrom
- well_height=1.0, altura de 1.0 eV
- N=2048, 2048 pontos na malha (discretização)
- dt=1e-19, passo de tempo de $1 \times 10^{-18}$ segundos
Calculo dos Autoestados e Autovalores¶
O comando abaxo solicita o cálculo dos autoestados e autovalores com precisao de 0.01 %.
Está precisão não é em relação ao valor analÃtico, mas em relação ao valor para o qual o algoritmo convergiu (que possui um erro em relação ao analÃtico).
In [6]:
poco_100.evolucao_imaginaria(precision=1e-4)
Energy [0]:
Numeric=3.4898948254e-03
Analytic=3.4828963796e-03
iterations=104000
Energy [1]:
Numeric=1.3951814274e-02
Analytic=1.3929748482e-02
iterations=76000
Energy [2]:
Numeric=3.1373388323e-02
Analytic=3.1334961728e-02
iterations=48000
Energy [3]:
Numeric=5.5750644769e-02
Analytic=5.5688922460e-02
iterations=41000
Energy [4]:
Numeric=8.7064283660e-02
Analytic=8.6977528175e-02
iterations=31000
Energy [5]:
Numeric=1.2528953568e-01
Analytic=1.2518144938e-01
iterations=29000
Energy [6]:
Numeric=1.7042458894e-01
Analytic=1.7027502006e-01
iterations=23000
Energy [7]:
Numeric=2.2240967245e-01
Analytic=2.2222458156e-01
iterations=22000
Energy [8]:
Numeric=2.8119865520e-01
Analytic=2.8098597641e-01
iterations=19000
Energy [9]:
Numeric=3.4675936894e-01
Analytic=3.4650064353e-01
iterations=18000
Energy [10]:
Numeric=4.1898720865e-01
Analytic=4.1868926581e-01
iterations=16000
Energy [11]:
Numeric=4.9778467623e-01
Analytic=4.9744080362e-01
iterations=15000
Energy [12]:
Numeric=5.8299325437e-01
Analytic=5.8259197003e-01
iterations=13000
Energy [13]:
Numeric=6.7429257248e-01
Analytic=6.7388421325e-01
iterations=13000
Energy [14]:
Numeric=7.7127282048e-01
Analytic=7.7085693460e-01
iterations=12000
Energy [15]:
Numeric=8.7295472920e-01
Analytic=8.7249408531e-01
iterations=12000
Energy [16]:
Numeric=9.7561504625e-01
Analytic=9.7487073823e-01
iterations=15000
Out[6]:
<__main__.PocoQuantico at 0x7fb7e1a47c18>
Como é possÃvel ver acima (clique na tela para rolar) temos 17 autoestados, com autovalores:
In [7]:
print(poco_100.eigenvalues)
[ 0.00348989 0.01395181 0.03137339 0.05575064 0.08706428 0.12528954 0.17042459 0.22240967 0.28119866 0.34675937 0.41898721 0.49778468 0.58299325 0.67429257 0.77127282 0.87295473 0.97561505]
Os autovalores analÃticos são:
In [8]:
poco_100.analytical_eigenvalues = np.array(poco_100.solucao_analitica())
print(poco_100.analytical_eigenvalues)
[ 0.0034829 0.01392975 0.03133496 0.05568892 0.08697753 0.12518145 0.17027502 0.22222458 0.28098598 0.34650064 0.41868927 0.4974408 0.58259197 0.67388421 0.77085693 0.87249409 0.97487074]
Vamos ver o erro em cada autovalor:
In [9]:
eigenvalues = pd.DataFrame({'analytic': poco_100.analytical_eigenvalues, 'numeric': poco_100.eigenvalues})
eigenvalues['error (%)'] = eigenvalues.apply(lambda l: 100.0*(l['numeric']/l['analytic']-1.0), axis=1)
print(eigenvalues)
analytic numeric error (%) 0 0.003483 0.003490 0.200938 1 0.013930 0.013952 0.158408 2 0.031335 0.031373 0.122632 3 0.055689 0.055751 0.110834 4 0.086978 0.087064 0.099745 5 0.125181 0.125290 0.086344 6 0.170275 0.170425 0.087840 7 0.222225 0.222410 0.083290 8 0.280986 0.281199 0.075690 9 0.346501 0.346759 0.074668 10 0.418689 0.418987 0.071161 11 0.497441 0.497785 0.069128 12 0.582592 0.582993 0.068879 13 0.673884 0.674293 0.060598 14 0.770857 0.771273 0.053951 15 0.872494 0.872955 0.052796 16 0.974871 0.975615 0.076349
Vamos imprimir os autoestados:
In [10]:
fix, axs = plt.subplots(5,4, figsize=(22,22))
for i in range(5):
for j in range(4):
if (4*i+j) >= len(poco_100.eigenstates):
continue
axs[i,j].plot(poco_100.z_ang, np.abs(poco_100.eigenstates[4*i+j])**2)
axs[i,j].set_xlabel(r'z (A)')
axs[i,j].set_ylabel(r'$|\psi_{%d} (z)|^2$' % (4*i+j))
plt.show()
Estes autoestados estão normalizados para um eixo z em unidades atômicas, vamos normalizar eles para um eixo z em Angstrom:
In [11]:
z = poco_100.z_ang
def normalize2ang(psi):
# psi / (<psi|psi>)^(1/2)
return psi / np.sqrt(simps(psi.conj()*psi,z))
poco_100.eigenstates_ang = [normalize2ang(es) for es in poco_100.eigenstates]
vamos imprimir para ver que está tudo certo:
In [12]:
fix, axs = plt.subplots(5,4, figsize=(22,22))
for i in range(5):
for j in range(4):
if (4*i+j) >= len(poco_100.eigenstates_ang):
continue
axs[i,j].plot(poco_100.z_ang, np.abs(poco_100.eigenstates_ang[4*i+j])**2)
axs[i,j].set_xlabel(r'z (A)')
axs[i,j].set_ylabel(r'$|\psi_{%d} (z)|^2$' % (4*i+j))
plt.show()
Vamos colocar os dados em um DataFrame para depois exportá-los para um .csv.
In [13]:
export_values = {
'z_ang': poco_100.z_ang, # o eixo z em Angstrom
'v_ev': poco_100.v_ev, # o potencial em eV
}
for i, state in enumerate(poco_100.eigenstates_ang):
export_values['psi_%02d' % i] = state # psi
export_values['psi_squared_%d' % i] = np.abs(state)**2 # |psi|^2
results = pd.DataFrame(export_values)
results.to_csv('saidas/poco_quantico_estados.csv')
results.head()
Out[13]:
| psi_00 | psi_01 | psi_02 | psi_03 | psi_04 | psi_05 | psi_06 | psi_07 | psi_08 | psi_09 | ... | psi_squared_2 | psi_squared_3 | psi_squared_4 | psi_squared_5 | psi_squared_6 | psi_squared_7 | psi_squared_8 | psi_squared_9 | v_ev | z_ang | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | (-4.82294817881e-15+1.12964096718e-15j) | (-3.29406817957e-12-9.23908767427e-17j) | (1.89685488369e-15-1.5486171481e-15j) | (-6.65684309279e-12+1.62156548197e-17j) | (2.58336684144e-15+1.71939908397e-15j) | (-1.00738141044e-11-1.08157349786e-15j) | (9.0381653665e-15-1.55288386129e-15j) | (-1.34261504582e-11+4.84283761882e-16j) | (1.25868856246e-14+7.83583350216e-16j) | (-1.67870529003e-11+2.10897374484e-16j) | ... | 5.996274e-30 | 4.431356e-23 | 9.630117e-30 | 1.014817e-22 | 8.409988e-29 | 1.802615e-22 | 1.590437e-28 | 2.818051e-22 | 1.0 | -150.000000 |
| 1 | (-1.05227960265e-14+1.03796143012e-15j) | (3.29422902649e-12-9.55842216151e-17j) | (-1.57371978487e-14-1.47538535868e-15j) | (6.65654736869e-12-3.23319078479e-17j) | (-2.68441416418e-14+1.59549638385e-15j) | (1.00753716049e-11-1.05032590027e-15j) | (-3.25773971547e-14-1.47869246396e-15j) | (1.34272094008e-11+4.88771433964e-16j) | (-4.09073678505e-14+7.12970155475e-16j) | (1.67897970222e-11+3.05038815012e-16j) | ... | 2.498362e-28 | 4.430962e-23 | 7.231535e-28 | 1.015131e-22 | 1.063473e-27 | 1.802900e-22 | 1.673921e-27 | 2.818973e-22 | 1.0 | -149.853444 |
| 2 | (1.22003432191e-15+9.5089444218e-16j) | (-3.29416290053e-12-9.01317914318e-17j) | (1.94546520686e-14-1.43497471378e-15j) | (-6.65692299997e-12-7.55440994737e-17j) | (3.23635863828e-14+1.50051379996e-15j) | (-1.00741501524e-11-9.7983700616e-16j) | (5.03772429289e-14-1.36916856347e-15j) | (-1.34267076408e-11+4.52422542553e-16j) | (6.63290640326e-14+6.35701013715e-16j) | (-1.67875923063e-11+3.61441494123e-16j) | ... | 3.805426e-28 | 4.431462e-23 | 1.049653e-27 | 1.014885e-22 | 2.539741e-27 | 1.802765e-22 | 4.399949e-27 | 2.818233e-22 | 1.0 | -149.706888 |
| 3 | (-1.60701395839e-14+8.92398761367e-16j) | (3.29439463925e-12-1.14971940227e-16j) | (-3.33998464234e-14-1.35396626355e-15j) | (6.65693848969e-12-9.10137357047e-17j) | (-5.618585553e-14+1.38946754031e-15j) | (1.00758083477e-11-9.39206819295e-16j) | (-7.4288299234e-14-1.32647489479e-15j) | (1.34277016281e-11+4.30129757794e-16j) | (-9.43444077801e-14+6.02188072166e-16j) | (1.67905319264e-11+3.77626964146e-16j) | ... | 1.117383e-27 | 4.431483e-23 | 3.158781e-27 | 1.015219e-22 | 5.520511e-27 | 1.803032e-22 | 8.901230e-27 | 2.819220e-22 | 1.0 | -149.560332 |
| 4 | (7.23442226821e-15+8.23075411143e-16j) | (-3.29435591684e-12-9.90009876382e-17j) | (3.70124492678e-14-1.31290057687e-15j) | (-6.65729237544e-12-8.88866409581e-17j) | (6.21056750103e-14+1.30348842182e-15j) | (-1.00748359522e-11-8.57115115791e-16j) | (9.17544563236e-14-1.21613583764e-15j) | (-1.34277087815e-11+3.82003362191e-16j) | (1.20033100157e-13+5.38664544098e-16j) | (-1.67887176089e-11+3.68970096635e-16j) | ... | 1.371645e-27 | 4.431954e-23 | 3.858814e-27 | 1.015023e-22 | 8.420359e-27 | 1.803034e-22 | 1.440824e-26 | 2.818610e-22 | 1.0 | -149.413776 |
5 rows × 36 columns
Nós temos também informações sobre tempo de processamento e número de iterações para cada nÃvel de energia:
In [14]:
perf_metrics = pd.DataFrame({
'time (s)': poco_100.timers,
'iterations': poco_100.counters,
'numeric precision (%)': 100*poco_100.eigenvalues_precisions,
'analytic (eV)': poco_100.analytical_eigenvalues,
'numeric (eV)': poco_100.eigenvalues,
'actual precision (%)': 100.0*(poco_100.eigenvalues/poco_100.analytical_eigenvalues-1.0)
})
perf_metrics.to_csv('saidas/poco_quantico_metricas.csv')
perf_metrics
Out[14]:
| actual precision (%) | analytic (eV) | iterations | numeric (eV) | numeric precision (%) | time (s) | |
|---|---|---|---|---|---|---|
| 0 | 0.200938 | 0.003483 | 104000 | 0.003490 | 0.009897 | 79.097671 |
| 1 | 0.158408 | 0.013930 | 76000 | 0.013952 | 0.009482 | 117.730071 |
| 2 | 0.122632 | 0.031335 | 48000 | 0.031373 | 0.008770 | 108.682453 |
| 3 | 0.110834 | 0.055689 | 41000 | 0.055751 | 0.009870 | 115.231905 |
| 4 | 0.099745 | 0.086978 | 31000 | 0.087064 | 0.009191 | 108.636536 |
| 5 | 0.086344 | 0.125181 | 29000 | 0.125290 | 0.007457 | 120.827942 |
| 6 | 0.087840 | 0.170275 | 23000 | 0.170425 | 0.009140 | 112.861944 |
| 7 | 0.083290 | 0.222225 | 22000 | 0.222410 | 0.007595 | 119.852372 |
| 8 | 0.075690 | 0.280986 | 19000 | 0.281199 | 0.006839 | 120.395632 |
| 9 | 0.074668 | 0.346501 | 18000 | 0.346759 | 0.006870 | 121.071307 |
| 10 | 0.071161 | 0.418689 | 16000 | 0.418987 | 0.006269 | 96.079016 |
| 11 | 0.069128 | 0.497441 | 15000 | 0.497785 | 0.007496 | 87.088524 |
| 12 | 0.068879 | 0.582592 | 13000 | 0.582993 | 0.009941 | 114.925910 |
| 13 | 0.060598 | 0.673884 | 13000 | 0.674293 | 0.007896 | 123.765117 |
| 14 | 0.053951 | 0.770857 | 12000 | 0.771273 | 0.008039 | 122.778017 |
| 15 | 0.052796 | 0.872494 | 12000 | 0.872955 | 0.008532 | 118.763190 |
| 16 | 0.076349 | 0.974871 | 15000 | 0.975615 | 0.008492 | 160.496706 |
Augumas relações interessantes¶
In [15]:
metrics = perf_metrics.copy()
metrics.reset_index(inplace=True)
ax = metrics[['iterations', 'time (s)']].plot(secondary_y='time (s)', figsize=(18,10))
ax.set_xlabel('eigenstate number')
ax.set_ylabel('Numero de iteracoes')
ax.right_ax.set_ylabel('Tempo de processamento (s)')
plt.show()
In [16]:
ax = metrics[['numeric precision (%)', 'actual precision (%)']].plot(secondary_y='actual precision (%)', figsize=(18,10))
ax.set_xlabel('eigenstate number')
ax.set_ylabel('Precisao Numerica (%)')
ax.right_ax.set_ylabel('Erro em relacao ao Valor Analitico (%)')
plt.show()
Verificando os resultados analíticos¶
Aqui vamos variar o dispositivo em largura e verificar os autovalores obtidos analiticamente.
In [17]:
eigenvalues = []
eigenvalues_per_length = []
for Vb in [1.0]:
for L in np.linspace(10, 80, 100):
d = PocoQuantico(well_length=L, well_height=Vb)
evs = d.solucao_analitica()
for ev in evs:
eigenvalues.append((L,ev))
eigenvalues_per_length.append((L, len(evs)))
print("finalizamos %.2f" % L)
finalizamos 10.00 finalizamos 10.71 finalizamos 11.41 finalizamos 12.12 finalizamos 12.83 finalizamos 13.54 finalizamos 14.24 finalizamos 14.95 finalizamos 15.66 finalizamos 16.36 finalizamos 17.07 finalizamos 17.78 finalizamos 18.48 finalizamos 19.19 finalizamos 19.90 finalizamos 20.61 finalizamos 21.31 finalizamos 22.02 finalizamos 22.73 finalizamos 23.43 finalizamos 24.14 finalizamos 24.85 finalizamos 25.56 finalizamos 26.26 finalizamos 26.97 finalizamos 27.68 finalizamos 28.38 finalizamos 29.09 finalizamos 29.80 finalizamos 30.51 finalizamos 31.21 finalizamos 31.92 finalizamos 32.63 finalizamos 33.33 finalizamos 34.04 finalizamos 34.75 finalizamos 35.45 finalizamos 36.16 finalizamos 36.87 finalizamos 37.58 finalizamos 38.28 finalizamos 38.99 finalizamos 39.70 finalizamos 40.40 finalizamos 41.11 finalizamos 41.82 finalizamos 42.53 finalizamos 43.23 finalizamos 43.94 finalizamos 44.65 finalizamos 45.35 finalizamos 46.06 finalizamos 46.77 finalizamos 47.47 finalizamos 48.18 finalizamos 48.89 finalizamos 49.60 finalizamos 50.30 finalizamos 51.01 finalizamos 51.72 finalizamos 52.42 finalizamos 53.13 finalizamos 53.84 finalizamos 54.55 finalizamos 55.25 finalizamos 55.96 finalizamos 56.67 finalizamos 57.37 finalizamos 58.08 finalizamos 58.79 finalizamos 59.49 finalizamos 60.20 finalizamos 60.91 finalizamos 61.62 finalizamos 62.32 finalizamos 63.03 finalizamos 63.74 finalizamos 64.44 finalizamos 65.15 finalizamos 65.86 finalizamos 66.57 finalizamos 67.27 finalizamos 67.98 finalizamos 68.69 finalizamos 69.39 finalizamos 70.10 finalizamos 70.81 finalizamos 71.52 finalizamos 72.22 finalizamos 72.93 finalizamos 73.64 finalizamos 74.34 finalizamos 75.05 finalizamos 75.76 finalizamos 76.46 finalizamos 77.17 finalizamos 77.88 finalizamos 78.59 finalizamos 79.29 finalizamos 80.00
In [18]:
fix, ax = plt.subplots(1,1, figsize=(18,14))
x,y=tuple(zip(*eigenvalues))
ax.scatter(x,y, marker='.')
plt.show()
In [19]:
fix, ax = plt.subplots(1,1, figsize=(18,10))
x,y=tuple(zip(*eigenvalues_per_length))
ax.scatter(x,y)
plt.show()
vamos exportar os valores
In [20]:
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_por_comprimento.csv')
#l_e
In [21]:
L,n=tuple(zip(*eigenvalues_per_length))
l_n = pd.DataFrame({'L':L, 'n':n})
l_n.to_csv('saidas/poco_quantico_numero_de_autovalores_por_comprimento.csv')
#l_n
In [22]:
eigenvalues = []
eigenvalues_per_length = []
for Vb in [1.0]:
for L in np.linspace(10, 80, 100):
d = PocoQuantico(well_length=L, well_height=Vb, N=2048, dt=1e-18)
d.evolucao_imaginaria(precision=1e-4)
evs = d.eigenvalues
for ev in evs:
eigenvalues.append((L,ev))
eigenvalues_per_length.append((L, len(evs)))
print('Estamos em L = %.2f' % L)
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv')
Energy [0]:
Numeric=1.9087770884e-01
Analytic=1.9076054414e-01
iterations=5000
Energy [1]:
Numeric=7.0451637873e-01
Analytic=7.0276957111e-01
iterations=5000
Estamos em L = 10.00
Energy [0]:
Numeric=1.7334442584e-01
Analytic=1.7322678903e-01
iterations=5000
Energy [1]:
Numeric=6.4824094252e-01
Analytic=6.4731212760e-01
iterations=5000
Estamos em L = 10.71
Energy [0]:
Numeric=1.5807630540e-01
Analytic=1.5796347786e-01
iterations=5000
Energy [1]:
Numeric=5.9741061257e-01
Analytic=5.9681360721e-01
iterations=5000
Estamos em L = 11.41
Energy [0]:
Numeric=1.4470971004e-01
Analytic=1.4460324525e-01
iterations=5000
Energy [1]:
Numeric=5.5155953861e-01
Analytic=5.5110359698e-01
iterations=5000
Estamos em L = 12.12
Energy [0]:
Numeric=1.3294818095e-01
Analytic=1.3284809547e-01
iterations=5000
Energy [1]:
Numeric=5.1022850498e-01
Analytic=5.0983916870e-01
iterations=5000
Energy [2]:
Numeric=9.7844069033e-01
Analytic=9.8552408029e-01
iterations=16000
Estamos em L = 12.83
Energy [0]:
Numeric=1.2254765693e-01
Analytic=1.2245482543e-01
iterations=6000
Energy [1]:
Numeric=4.7296750814e-01
Analytic=4.7261538262e-01
iterations=5000
Energy [2]:
Numeric=9.4311265450e-01
Analytic=9.4695789904e-01
iterations=14000
Estamos em L = 13.54
Energy [0]:
Numeric=1.1331069022e-01
Analytic=1.1322372030e-01
iterations=6000
Energy [1]:
Numeric=4.3934810356e-01
Analytic=4.3902109138e-01
iterations=5000
Energy [2]:
Numeric=8.9951495650e-01
Analytic=9.0062555928e-01
iterations=11000
Estamos em L = 14.24
Energy [0]:
Numeric=1.0507156265e-01
Analytic=1.0498977229e-01
iterations=6000
Energy [1]:
Numeric=4.0896879556e-01
Analytic=4.0866592077e-01
iterations=6000
Energy [2]:
Numeric=8.5249581616e-01
Analytic=8.5250675297e-01
iterations=9000
Estamos em L = 14.95
Energy [0]:
Numeric=9.7692079919e-02
Analytic=9.7615835088e-02
iterations=7000
Energy [1]:
Numeric=3.8147770720e-01
Analytic=3.8119200267e-01
iterations=6000
Energy [2]:
Numeric=8.0552166579e-01
Analytic=8.0517558023e-01
iterations=8000
Estamos em L = 15.66
Energy [0]:
Numeric=9.1059086969e-02
Analytic=9.0987262182e-02
iterations=7000
Energy [1]:
Numeric=3.5654792794e-01
Analytic=3.5627770760e-01
iterations=6000
Energy [2]:
Numeric=7.6027571508e-01
Analytic=7.5981577825e-01
iterations=7000
Estamos em L = 16.36
Energy [0]:
Numeric=8.5075688988e-02
Analytic=8.5007685233e-02
iterations=7000
Energy [1]:
Numeric=3.3389335704e-01
Analytic=3.3363729016e-01
iterations=6000
Energy [2]:
Numeric=7.1744415556e-01
Analytic=7.1695055514e-01
iterations=6000
Estamos em L = 17.07
Energy [0]:
Numeric=7.9660580891e-02
Analytic=7.9595671364e-02
iterations=7000
Energy [1]:
Numeric=3.1326171735e-01
Analytic=3.1301857022e-01
iterations=6000
Energy [2]:
Numeric=6.7724417228e-01
Analytic=6.7676474498e-01
iterations=6000
Estamos em L = 17.78
Energy [0]:
Numeric=7.4742420431e-02
Analytic=7.4682061157e-02
iterations=8000
Energy [1]:
Numeric=2.9443130471e-01
Analytic=2.9419979640e-01
iterations=6000
Energy [2]:
Numeric=6.3972772356e-01
Analytic=6.3926404915e-01
iterations=6000
Energy [3]:
Numeric=1.0146771640e+00
Analytic=9.9953527480e-01
iterations=14000
Estamos em L = 18.48
Energy [0]:
Numeric=7.0265462997e-02
Analytic=7.0207836528e-02
iterations=8000
Energy [1]:
Numeric=2.7720428753e-01
Analytic=2.7698630086e-01
iterations=7000
Energy [2]:
Numeric=6.0480494101e-01
Analytic=6.0435942918e-01
iterations=6000
Energy [3]:
Numeric=9.8172408479e-01
Analytic=9.7791008155e-01
iterations=13000
Estamos em L = 19.19
Energy [0]:
Numeric=6.6176393151e-02
Analytic=6.6122403312e-02
iterations=9000
Energy [1]:
Numeric=2.6141472421e-01
Analytic=2.6120725666e-01
iterations=7000
Energy [2]:
Numeric=5.7234233252e-01
Analytic=5.7191381150e-01
iterations=6000
Energy [3]:
Numeric=9.4516355070e-01
Analytic=9.4374687345e-01
iterations=11000
Estamos em L = 19.90
Energy [0]:
Numeric=6.2433831333e-02
Analytic=6.2382200205e-02
iterations=9000
Energy [1]:
Numeric=2.4691062584e-01
Analytic=2.4671268467e-01
iterations=7000
Energy [2]:
Numeric=5.4217948172e-01
Analytic=5.4176858088e-01
iterations=6000
Energy [3]:
Numeric=9.0677023073e-01
Analytic=9.0595404030e-01
iterations=10000
Estamos em L = 20.61
Energy [0]:
Numeric=5.8999372120e-02
Analytic=5.8949565955e-02
iterations=9000
Energy [1]:
Numeric=2.3356030754e-01
Analytic=2.3337076703e-01
iterations=7000
Energy [2]:
Numeric=5.1415230715e-01
Analytic=5.1375874376e-01
iterations=6000
Energy [3]:
Numeric=8.6814979351e-01
Analytic=8.6749062249e-01
iterations=9000
Estamos em L = 21.31
Energy [0]:
Numeric=5.5838399930e-02
Analytic=5.5791812061e-02
iterations=10000
Energy [1]:
Numeric=2.2124440237e-01
Analytic=2.2106547621e-01
iterations=8000
Energy [2]:
Numeric=4.8810007366e-01
Analytic=4.8772154005e-01
iterations=6000
Energy [3]:
Numeric=8.3027603352e-01
Analytic=8.2965954663e-01
iterations=8000
Estamos em L = 22.02
Energy [0]:
Numeric=5.2925470487e-02
Analytic=5.2880459850e-02
iterations=10000
Energy [1]:
Numeric=2.0986572187e-01
Analytic=2.0969450457e-01
iterations=8000
Energy [2]:
Numeric=4.6386405273e-01
Analytic=4.6350118696e-01
iterations=6000
Energy [3]:
Numeric=7.9369415684e-01
Analytic=7.9308939304e-01
iterations=7000
Estamos em L = 22.73
Energy [0]:
Numeric=5.0232902498e-02
Analytic=5.0190609763e-02
iterations=11000
Energy [1]:
Numeric=1.9933191304e-01
Analytic=1.9916746783e-01
iterations=8000
Energy [2]:
Numeric=4.4130071791e-01
Analytic=4.4095131236e-01
iterations=6000
Energy [3]:
Numeric=7.5865051612e-01
Analytic=7.5808247621e-01
iterations=7000
Estamos em L = 23.43
Energy [0]:
Numeric=4.7741373832e-02
Analytic=4.7700417507e-02
iterations=11000
Energy [1]:
Numeric=1.8956316286e-01
Analytic=1.8940435241e-01
iterations=8000
Energy [2]:
Numeric=4.2027173473e-01
Analytic=4.1993600366e-01
iterations=6000
Energy [3]:
Numeric=7.2530969843e-01
Analytic=7.2476781838e-01
iterations=7000
Estamos em L = 24.14
Energy [0]:
Numeric=4.5430763809e-02
Analytic=4.5390657015e-02
iterations=11000
Energy [1]:
Numeric=1.8048374270e-01
Analytic=1.8033417623e-01
iterations=9000
Energy [2]:
Numeric=4.0065288109e-01
Analytic=4.0033003326e-01
iterations=6000
Energy [3]:
Numeric=6.9370188194e-01
Analytic=6.9317757857e-01
iterations=7000
Energy [4]:
Numeric=9.9406640139e-01
Analytic=9.9545379356e-01
iterations=33000
Estamos em L = 24.85
Energy [0]:
Numeric=4.3281847119e-02
Analytic=4.3244354258e-02
iterations=12000
Energy [1]:
Numeric=1.7203803344e-01
Analytic=1.7189383463e-01
iterations=9000
Energy [2]:
Numeric=3.8232528318e-01
Analytic=3.8201860448e-01
iterations=7000
Energy [3]:
Numeric=6.6379388157e-01
Analytic=6.6328855024e-01
iterations=7000
Energy [4]:
Numeric=9.7190460266e-01
Analytic=9.7158004037e-01
iterations=19000
Estamos em L = 25.56
Energy [0]:
Numeric=4.1283290383e-02
Analytic=4.1246479170e-02
iterations=12000
Energy [1]:
Numeric=1.6416694982e-01
Analytic=1.6402710597e-01
iterations=9000
Energy [2]:
Numeric=3.6519110117e-01
Analytic=3.6489683037e-01
iterations=7000
Energy [3]:
Numeric=6.3553282516e-01
Analytic=6.3504598865e-01
iterations=7000
Energy [4]:
Numeric=9.4214178018e-01
Analytic=9.4146885537e-01
iterations=13000
Estamos em L = 26.26
Energy [0]:
Numeric=3.9418200905e-02
Analytic=3.9383685421e-02
iterations=13000
Energy [1]:
Numeric=1.5681568181e-01
Analytic=1.5668379406e-01
iterations=10000
Energy [2]:
Numeric=3.4915353236e-01
Analytic=3.4886907686e-01
iterations=7000
Energy [3]:
Numeric=6.0885088613e-01
Analytic=6.0837781652e-01
iterations=7000
Energy [4]:
Numeric=9.1025777369e-01
Analytic=9.0954836051e-01
iterations=10000
Estamos em L = 26.97
Energy [0]:
Numeric=3.7678091628e-02
Analytic=3.7644089781e-02
iterations=13000
Energy [1]:
Numeric=1.4994671853e-01
Analytic=1.4981898803e-01
iterations=10000
Energy [2]:
Numeric=3.3412217372e-01
Analytic=3.3384825084e-01
iterations=7000
Energy [3]:
Numeric=5.8365984090e-01
Analytic=5.8320329872e-01
iterations=7000
Energy [4]:
Numeric=8.7807862853e-01
Analytic=8.7741382756e-01
iterations=9000
Estamos em L = 27.68
Energy [0]:
Numeric=3.6049031396e-02
Analytic=3.6017084372e-02
iterations=14000
Energy [1]:
Numeric=1.4351694771e-01
Analytic=1.4339242258e-01
iterations=10000
Energy [2]:
Numeric=3.2002116991e-01
Analytic=3.1975508218e-01
iterations=7000
Energy [3]:
Numeric=5.5987849867e-01
Analytic=5.5943840624e-01
iterations=7000
Energy [4]:
Numeric=8.4644803277e-01
Analytic=8.4580128982e-01
iterations=8000
Estamos em L = 28.38
Energy [0]:
Numeric=3.4524773766e-02
Analytic=3.4493176330e-02
iterations=14000
Energy [1]:
Numeric=1.3748532249e-01
Analytic=1.3736792409e-01
iterations=11000
Energy [2]:
Numeric=3.0677529995e-01
Analytic=3.0651742841e-01
iterations=7000
Energy [3]:
Numeric=5.3742777231e-01
Analytic=5.3699913749e-01
iterations=7000
Energy [4]:
Numeric=8.1568933177e-01
Analytic=8.1508262046e-01
iterations=8000
Estamos em L = 29.09
Energy [0]:
Numeric=3.3093579193e-02
Analytic=3.3063850441e-02
iterations=15000
Energy [1]:
Numeric=1.3182723909e-01
Analytic=1.3171293033e-01
iterations=11000
Energy [2]:
Numeric=2.9431330009e-01
Analytic=2.9406961812e-01
iterations=8000
Energy [3]:
Numeric=5.1621786570e-01
Analytic=5.1580355070e-01
iterations=7000
Energy [4]:
Numeric=7.8605944110e-01
Analytic=7.8544779063e-01
iterations=7000
Estamos em L = 29.80
Energy [0]:
Numeric=3.1750989414e-02
Analytic=3.1721451053e-02
iterations=15000
Energy [1]:
Numeric=1.2651022724e-01
Analytic=1.2639807314e-01
iterations=11000
Energy [2]:
Numeric=2.8258699575e-01
Analytic=2.8235184108e-01
iterations=8000
Energy [3]:
Numeric=4.9617331541e-01
Analytic=4.9577297042e-01
iterations=7000
Energy [4]:
Numeric=7.5757186815e-01
Analytic=7.5698691492e-01
iterations=7000
Estamos em L = 30.51
Energy [0]:
Numeric=3.0486895326e-02
Analytic=3.0459080254e-02
iterations=16000
Energy [1]:
Numeric=1.2150234495e-01
Analytic=1.2139681506e-01
iterations=12000
Energy [2]:
Numeric=2.7153873174e-01
Analytic=2.7130958797e-01
iterations=8000
Energy [3]:
Numeric=4.7722491952e-01
Analytic=4.7683265955e-01
iterations=7000
Energy [4]:
Numeric=7.3029517181e-01
Analytic=7.2973211299e-01
iterations=7000
Energy [5]:
Numeric=9.9182046597e-01
Analytic=9.8988701840e-01
iterations=20000
Estamos em L = 31.21
Energy [0]:
Numeric=2.9298289326e-02
Analytic=2.9270509818e-02
iterations=16000
Energy [1]:
Numeric=1.1678854657e-01
Analytic=1.1668513249e-01
iterations=12000
Energy [2]:
Numeric=2.6111571174e-01
Analytic=2.6089313978e-01
iterations=8000
Energy [3]:
Numeric=4.5928231352e-01
Analytic=4.5891214420e-01
iterations=8000
Energy [4]:
Numeric=7.0422395169e-01
Analytic=7.0368074793e-01
iterations=7000
Energy [5]:
Numeric=9.6746384556e-01
Analytic=9.6647148151e-01
iterations=15000
Estamos em L = 31.92
Energy [0]:
Numeric=2.8176275003e-02
Analytic=2.8150104830e-02
iterations=17000
Energy [1]:
Numeric=1.1233909363e-01
Analytic=1.1224123869e-01
iterations=13000
Energy [2]:
Numeric=2.5126789045e-01
Analytic=2.5105710490e-01
iterations=9000
Energy [3]:
Numeric=4.4230641208e-01
Analytic=4.4194531377e-01
iterations=8000
Energy [4]:
Numeric=6.7934119465e-01
Analytic=6.7880911221e-01
iterations=7000
Energy [5]:
Numeric=9.4059362866e-01
Analytic=9.3977572157e-01
iterations=12000
Estamos em L = 32.63
Energy [0]:
Numeric=2.7117585792e-02
Analytic=2.7092757257e-02
iterations=18000
Energy [1]:
Numeric=1.0814121233e-01
Analytic=1.0804534120e-01
iterations=13000
Energy [2]:
Numeric=2.4196535570e-01
Analytic=2.4176000106e-01
iterations=9000
Energy [3]:
Numeric=4.2621993014e-01
Analytic=4.2587037705e-01
iterations=8000
Energy [4]:
Numeric=6.5559671092e-01
Analytic=6.5508083497e-01
iterations=7000
Energy [5]:
Numeric=9.1291173575e-01
Analytic=9.1214425039e-01
iterations=10000
Estamos em L = 33.33
Energy [0]:
Numeric=2.6118593760e-02
Analytic=2.6093828026e-02
iterations=18000
Energy [1]:
Numeric=1.0417410572e-01
Analytic=1.0407942900e-01
iterations=13000
Energy [2]:
Numeric=2.3316341076e-01
Analytic=2.3296387866e-01
iterations=9000
Energy [3]:
Numeric=4.1096815544e-01
Analytic=4.1062972816e-01
iterations=8000
Energy [4]:
Numeric=6.3295225116e-01
Analytic=6.3245220289e-01
iterations=7000
Energy [5]:
Numeric=8.8522294590e-01
Analytic=8.8450302434e-01
iterations=9000
Estamos em L = 34.04
Energy [0]:
Numeric=2.5172589807e-02
Analytic=2.5149096376e-02
iterations=19000
Energy [1]:
Numeric=1.0041634792e-01
Analytic=1.0032708535e-01
iterations=14000
Energy [2]:
Numeric=2.2482987681e-01
Analytic=2.2463398216e-01
iterations=9000
Energy [3]:
Numeric=3.9650261151e-01
Analytic=3.9616975846e-01
iterations=8000
Energy [4]:
Numeric=6.1136003871e-01
Analytic=6.1087559633e-01
iterations=7000
Energy [5]:
Numeric=8.5795723943e-01
Analytic=8.5730108809e-01
iterations=9000
Estamos em L = 34.75
Energy [0]:
Numeric=2.4278294743e-02
Analytic=2.4254715466e-02
iterations=19000
Energy [1]:
Numeric=9.6861452283e-02
Analytic=9.6773322857e-02
iterations=14000
Energy [2]:
Numeric=2.1692247757e-01
Analytic=2.1673844569e-01
iterations=10000
Energy [3]:
Numeric=3.8275460241e-01
Analytic=3.8244063851e-01
iterations=9000
Energy [4]:
Numeric=5.9077858500e-01
Analytic=5.9030173471e-01
iterations=7000
Energy [5]:
Numeric=8.3143702722e-01
Analytic=8.3077498245e-01
iterations=8000
Estamos em L = 35.45
Energy [0]:
Numeric=2.3429533971e-02
Analytic=2.3407173382e-02
iterations=20000
Energy [1]:
Numeric=9.3487854064e-02
Analytic=9.3404437965e-02
iterations=15000
Energy [2]:
Numeric=2.0942813744e-01
Analytic=2.0924801994e-01
iterations=10000
Energy [3]:
Numeric=3.6970358729e-01
Analytic=3.6939608633e-01
iterations=9000
Energy [4]:
Numeric=5.7114506152e-01
Analytic=5.7068114937e-01
iterations=7000
Energy [5]:
Numeric=8.0568020243e-01
Analytic=8.0505134347e-01
iterations=8000
Estamos em L = 36.16
Energy [0]:
Numeric=2.2625851733e-02
Analytic=2.2603258788e-02
iterations=20000
Energy [1]:
Numeric=9.0290273691e-02
Analytic=9.0207882209e-02
iterations=15000
Energy [2]:
Numeric=2.0231139839e-01
Analytic=2.0213582686e-01
iterations=10000
Energy [3]:
Numeric=3.5729153806e-01
Analytic=3.5699313243e-01
iterations=9000
Energy [4]:
Numeric=5.5241663342e-01
Analytic=5.5196514500e-01
iterations=7000
Energy [5]:
Numeric=7.8079879270e-01
Analytic=7.8019462235e-01
iterations=8000
Energy [6]:
Numeric=1.0007108711e+00
Analytic=9.9967865883e-01
iterations=46000
Estamos em L = 36.87
Energy [0]:
Numeric=2.1861445134e-02
Analytic=2.1840030615e-02
iterations=21000
Energy [1]:
Numeric=8.7254249977e-02
Analytic=8.7172148158e-02
iterations=15000
Energy [2]:
Numeric=1.9555047071e-01
Analytic=1.9537713943e-01
iterations=10000
Energy [3]:
Numeric=3.4548163247e-01
Analytic=3.4519188876e-01
iterations=9000
Energy [4]:
Numeric=5.3453184241e-01
Analytic=5.3410641695e-01
iterations=8000
Energy [5]:
Numeric=7.5681535033e-01
Analytic=7.5623203684e-01
iterations=8000
Energy [6]:
Numeric=9.8554730482e-01
Analytic=9.8440374092e-01
iterations=20000
Estamos em L = 37.58
Energy [0]:
Numeric=2.1135190639e-02
Analytic=2.1114791258e-02
iterations=22000
Energy [1]:
Numeric=8.4363986435e-02
Analytic=8.4286668167e-02
iterations=16000
Energy [2]:
Numeric=1.8911269720e-01
Analytic=1.8894918387e-01
iterations=11000
Energy [3]:
Numeric=3.3424219049e-01
Analytic=3.3395532590e-01
iterations=9000
Energy [4]:
Numeric=5.1747895196e-01
Analytic=5.1705943464e-01
iterations=8000
Energy [5]:
Numeric=7.3373180018e-01
Analytic=7.3316772113e-01
iterations=8000
Energy [6]:
Numeric=9.6327832643e-01
Analytic=9.6234527513e-01
iterations=14000
Estamos em L = 38.28
Energy [0]:
Numeric=2.0445705779e-02
Analytic=2.0425062814e-02
iterations=22000
Energy [1]:
Numeric=8.1618807972e-02
Analytic=8.1541724340e-02
iterations=16000
Energy [2]:
Numeric=1.8299051111e-01
Analytic=1.8283096193e-01
iterations=11000
Energy [3]:
Numeric=3.2351789784e-01
Analytic=3.2324906098e-01
iterations=10000
Energy [4]:
Numeric=5.0118818006e-01
Analytic=5.0078066477e-01
iterations=8000
Energy [5]:
Numeric=7.1154855784e-01
Analytic=7.1099123458e-01
iterations=8000
Energy [6]:
Numeric=9.3925793974e-01
Analytic=9.3847063393e-01
iterations=12000
Estamos em L = 38.99
Energy [0]:
Numeric=1.9788218243e-02
Analytic=1.9768565975e-02
iterations=23000
Energy [1]:
Numeric=7.9001174616e-02
Analytic=7.8928368318e-02
iterations=17000
Energy [2]:
Numeric=1.7716066362e-01
Analytic=1.7700309103e-01
iterations=11000
Energy [3]:
Numeric=3.1330610143e-01
Analytic=3.1304115803e-01
iterations=10000
Energy [4]:
Numeric=4.8562450564e-01
Analytic=4.8522868448e-01
iterations=8000
Energy [5]:
Numeric=6.9022491809e-01
Analytic=6.8968290423e-01
iterations=8000
Energy [6]:
Numeric=9.1487109024e-01
Analytic=9.1411019055e-01
iterations=10000
Estamos em L = 39.70
Energy [0]:
Numeric=1.9161984770e-02
Analytic=1.9143201251e-02
iterations=24000
Energy [1]:
Numeric=7.6511007915e-02
Analytic=7.6438349704e-02
iterations=17000
Energy [2]:
Numeric=1.7159683943e-01
Analytic=1.7144766043e-01
iterations=12000
Energy [3]:
Numeric=3.0355991219e-01
Analytic=3.0330194120e-01
iterations=10000
Energy [4]:
Numeric=4.7075572104e-01
Analytic=4.7036421916e-01
iterations=8000
Energy [5]:
Numeric=6.6974422829e-01
Analytic=6.6921728858e-01
iterations=8000
Energy [6]:
Numeric=8.9055968331e-01
Analytic=8.8983488476e-01
iterations=9000
Estamos em L = 40.40
Energy [0]:
Numeric=1.8566093493e-02
Analytic=1.8547032232e-02
iterations=24000
Energy [1]:
Numeric=7.4132830098e-02
Analytic=7.4064052081e-02
iterations=18000
Energy [2]:
Numeric=1.6629384660e-01
Analytic=1.6614810166e-01
iterations=12000
Energy [3]:
Numeric=2.9425969700e-01
Analytic=2.9400382094e-01
iterations=10000
Energy [4]:
Numeric=4.5653246334e-01
Analytic=4.5615012889e-01
iterations=8000
Energy [5]:
Numeric=6.5007741851e-01
Analytic=6.4956547665e-01
iterations=8000
Energy [6]:
Numeric=8.6660065964e-01
Analytic=8.6593483383e-01
iterations=9000
Estamos em L = 41.11
Energy [0]:
Numeric=1.7996477679e-02
Analytic=1.7978270632e-02
iterations=25000
Energy [1]:
Numeric=7.1867171475e-02
Analytic=7.1798435728e-02
iterations=18000
Energy [2]:
Numeric=1.6123316722e-01
Analytic=1.6108907172e-01
iterations=12000
Energy [3]:
Numeric=2.8536094491e-01
Analytic=2.8512113299e-01
iterations=11000
Energy [4]:
Numeric=4.4291202063e-01
Analytic=4.4255136060e-01
iterations=9000
Energy [5]:
Numeric=6.3120488071e-01
Analytic=6.3069663909e-01
iterations=8000
Energy [6]:
Numeric=8.4324692185e-01
Analytic=8.4256862376e-01
iterations=8000
Estamos em L = 41.82
Energy [0]:
Numeric=1.7452711297e-02
Analytic=1.7435262914e-02
iterations=26000
Energy [1]:
Numeric=6.9700159502e-02
Analytic=6.9634986246e-02
iterations=19000
Energy [2]:
Numeric=1.5639944044e-01
Analytic=1.5625634767e-01
iterations=12000
Energy [3]:
Numeric=2.7686690721e-01
Analytic=2.7662998955e-01
iterations=11000
Energy [4]:
Numeric=4.2988461669e-01
Analytic=4.2953487775e-01
iterations=9000
Energy [5]:
Numeric=6.1307530430e-01
Analytic=6.1257908567e-01
iterations=8000
Energy [6]:
Numeric=8.2047024109e-01
Analytic=8.1982446342e-01
iterations=8000
Estamos em L = 42.53
Energy [0]:
Numeric=1.6934251371e-02
Analytic=1.6916478287e-02
iterations=26000
Energy [1]:
Numeric=6.7632917205e-02
Analytic=6.7567668421e-02
iterations=19000
Energy [2]:
Numeric=1.5177049222e-01
Analytic=1.5163673141e-01
iterations=13000
Energy [3]:
Numeric=2.6873949307e-01
Analytic=2.6850814208e-01
iterations=11000
Energy [4]:
Numeric=4.1741653792e-01
Analytic=4.1706957599e-01
iterations=9000
Energy [5]:
Numeric=5.9566557600e-01
Analytic=5.9518098902e-01
iterations=8000
Energy [6]:
Numeric=7.9837115452e-01
Analytic=7.9774962537e-01
iterations=8000
Energy [7]:
Numeric=9.9863540089e-01
Analytic=9.9648897578e-01
iterations=26000
Estamos em L = 43.23
Energy [0]:
Numeric=1.6437520152e-02
Analytic=1.6420497957e-02
iterations=27000
Energy [1]:
Numeric=6.5652828861e-02
Analytic=6.5590884714e-02
iterations=20000
Energy [2]:
Numeric=1.4735041929e-01
Analytic=1.4721796358e-01
iterations=13000
Energy [3]:
Numeric=2.6096506735e-01
Analytic=2.6073485502e-01
iterations=11000
Energy [4]:
Numeric=4.0546445839e-01
Analytic=4.0512619087e-01
iterations=9000
Energy [5]:
Numeric=5.7892662078e-01
Analytic=5.7847087910e-01
iterations=9000
Energy [6]:
Numeric=7.7696762329e-01
Analytic=7.7636623138e-01
iterations=8000
Energy [7]:
Numeric=9.8057125492e-01
Analytic=9.7947490366e-01
iterations=18000
Estamos em L = 43.94
Energy [0]:
Numeric=1.5962354453e-02
Analytic=1.5946005431e-02
iterations=28000
Energy [1]:
Numeric=6.3761582961e-02
Analytic=6.3699437872e-02
iterations=20000
Energy [2]:
Numeric=1.4312037534e-01
Analytic=1.4298864562e-01
iterations=13000
Energy [3]:
Numeric=2.5350607035e-01
Analytic=2.5329078962e-01
iterations=12000
Energy [4]:
Numeric=3.9400719151e-01
Analytic=3.9367720169e-01
iterations=9000
Energy [5]:
Numeric=5.6287102485e-01
Analytic=5.6241797749e-01
iterations=9000
Energy [6]:
Numeric=7.5626373975e-01
Analytic=7.5568038998e-01
iterations=8000
Energy [7]:
Numeric=9.5986840953e-01
Analytic=9.5897687552e-01
iterations=14000
Estamos em L = 44.65
Energy [0]:
Numeric=1.5508506200e-02
Analytic=1.5491777807e-02
iterations=28000
Energy [1]:
Numeric=6.1947549085e-02
Analytic=6.1888497189e-02
iterations=21000
Energy [2]:
Numeric=1.3906146720e-01
Analytic=7.3124284724e-02
iterations=14000
Energy [3]:
Numeric=2.4637132106e-01
Analytic=1.3893816914e-01
iterations=12000
Energy [4]:
Numeric=3.8302616688e-01
Analytic=2.4615789725e-01
iterations=9000
Energy [5]:
Numeric=5.4743403625e-01
Analytic=3.8269673455e-01
iterations=9000
Energy [6]:
Numeric=7.3625421168e-01
Analytic=5.4699241886e-01
iterations=8000
Energy [7]:
Numeric=9.3823009558e-01
Analytic=7.3568779316e-01
iterations=12000
Energy [8]:
Numeric=1.0027410067e+00
Analytic=9.3743516691e-01
iterations=64000
Estamos em L = 45.35
Energy [0]:
Numeric=1.5072736001e-02
Analytic=1.5056677900e-02
iterations=29000
Energy [1]:
Numeric=6.0212951305e-02
Analytic=6.0153568030e-02
iterations=21000
Energy [2]:
Numeric=1.3517898585e-01
Analytic=1.3505665184e-01
iterations=14000
Energy [3]:
Numeric=2.3952837567e-01
Analytic=2.3931932143e-01
iterations=12000
Energy [4]:
Numeric=3.7246936587e-01
Analytic=3.7216046651e-01
iterations=10000
Energy [5]:
Numeric=5.3259575620e-01
Analytic=5.3216539209e-01
iterations=9000
Energy [6]:
Numeric=7.1694187139e-01
Analytic=7.1637729409e-01
iterations=8000
Energy [7]:
Numeric=9.1638131344e-01
Analytic=9.1565218532e-01
iterations=11000
Estamos em L = 46.06
Energy [0]:
Numeric=1.4655100060e-02
Analytic=1.4639647142e-02
iterations=30000
Energy [1]:
Numeric=5.8546928904e-02
Analytic=5.8490464255e-02
iterations=22000
Energy [2]:
Numeric=1.3145677726e-01
Analytic=1.3133487933e-01
iterations=14000
Energy [3]:
Numeric=2.3296822464e-01
Analytic=2.3275930779e-01
iterations=12000
Energy [4]:
Numeric=3.6234624345e-01
Analytic=3.6204553239e-01
iterations=10000
Energy [5]:
Numeric=5.1832852142e-01
Analytic=5.1790922411e-01
iterations=9000
Energy [6]:
Numeric=6.9828463248e-01
Analytic=6.9773326963e-01
iterations=8000
Energy [7]:
Numeric=8.9469999291e-01
Analytic=8.9399772467e-01
iterations=10000
Estamos em L = 46.77
Energy [0]:
Numeric=1.4255591197e-02
Analytic=1.4239699157e-02
iterations=30000
Energy [1]:
Numeric=5.6952213213e-02
Analytic=5.6895283235e-02
iterations=22000
Energy [2]:
Numeric=1.2787837636e-01
Analytic=1.2776425225e-01
iterations=15000
Energy [3]:
Numeric=2.2665768862e-01
Analytic=2.2646312154e-01
iterations=13000
Energy [4]:
Numeric=3.5263009146e-01
Analytic=3.5233043495e-01
iterations=10000
Energy [5]:
Numeric=5.0461765865e-01
Analytic=5.0419742271e-01
iterations=9000
Energy [6]:
Numeric=6.8027584792e-01
Analytic=6.7973721984e-01
iterations=8000
Energy [7]:
Numeric=8.7336440556e-01
Analytic=8.7266776108e-01
iterations=9000
Estamos em L = 47.47
Energy [0]:
Numeric=1.3871199984e-02
Analytic=1.3855913922e-02
iterations=31000
Energy [1]:
Numeric=5.5418539188e-02
Analytic=5.5364383194e-02
iterations=23000
Energy [2]:
Numeric=1.2445022740e-01
Analytic=1.2433673811e-01
iterations=15000
Energy [3]:
Numeric=2.2061070526e-01
Analytic=2.2041697158e-01
iterations=13000
Energy [4]:
Numeric=3.4328847067e-01
Analytic=3.4299495923e-01
iterations=10000
Energy [5]:
Numeric=4.9140048822e-01
Analytic=4.9100469001e-01
iterations=10000
Energy [6]:
Numeric=6.6289504988e-01
Analytic=6.6236887112e-01
iterations=8000
Energy [7]:
Numeric=8.5244250300e-01
Analytic=8.5177298333e-01
iterations=9000
Estamos em L = 48.18
Energy [0]:
Numeric=1.3502168463e-02
Analytic=1.3487432507e-02
iterations=32000
Energy [1]:
Numeric=5.3949120562e-02
Analytic=5.3894362618e-02
iterations=23000
Energy [2]:
Numeric=1.2115817062e-01
Analytic=1.2104482746e-01
iterations=15000
Energy [3]:
Numeric=2.1479834608e-01
Analytic=2.1460794088e-01
iterations=13000
Energy [4]:
Numeric=3.3430794553e-01
Analytic=3.3402009106e-01
iterations=10000
Energy [5]:
Numeric=4.7869193454e-01
Analytic=4.7830691519e-01
iterations=10000
Energy [6]:
Numeric=6.4612094643e-01
Analytic=6.4560694724e-01
iterations=8000
Energy [7]:
Numeric=8.3201856451e-01
Analytic=8.3137696747e-01
iterations=9000
Estamos em L = 48.89
Energy [0]:
Numeric=1.3147685559e-02
Analytic=1.3133452252e-02
iterations=33000
Energy [1]:
Numeric=5.2534145435e-02
Analytic=5.2482041542e-02
iterations=24000
Energy [2]:
Numeric=1.1798853920e-01
Analytic=1.1788149388e-01
iterations=16000
Energy [3]:
Numeric=2.0920489574e-01
Analytic=2.0902392252e-01
iterations=14000
Energy [4]:
Numeric=3.2566252369e-01
Analytic=3.2538794027e-01
iterations=11000
Energy [5]:
Numeric=4.6646568837e-01
Analytic=4.6608115259e-01
iterations=10000
Energy [6]:
Numeric=6.2992535276e-01
Analytic=6.2942971292e-01
iterations=9000
Energy [7]:
Numeric=8.1213512907e-01
Analytic=8.1151518369e-01
iterations=9000
Energy [8]:
Numeric=9.9379612289e-01
Analytic=9.9220231269e-01
iterations=22000
Estamos em L = 49.60
Energy [0]:
Numeric=1.2807907194e-02
Analytic=1.2793222420e-02
iterations=33000
Energy [1]:
Numeric=5.1177289210e-02
Analytic=5.1124444499e-02
iterations=24000
Energy [2]:
Numeric=1.1494589093e-01
Analytic=1.1484015749e-01
iterations=16000
Energy [3]:
Numeric=2.0383066272e-01
Analytic=2.0365356087e-01
iterations=14000
Energy [4]:
Numeric=3.1735002466e-01
Analytic=3.1708166817e-01
iterations=11000
Energy [5]:
Numeric=4.5468176850e-01
Analytic=4.5430558976e-01
iterations=10000
Energy [6]:
Numeric=6.1429935812e-01
Analytic=6.1381535854e-01
iterations=9000
Energy [7]:
Numeric=7.9280687796e-01
Analytic=7.9220544068e-01
iterations=9000
Energy [8]:
Numeric=9.7622249635e-01
Analytic=9.7517443620e-01
iterations=16000
Estamos em L = 50.30
Energy [0]:
Numeric=1.2480220160e-02
Analytic=1.2466040204e-02
iterations=34000
Energy [1]:
Numeric=4.9869074855e-02
Analytic=4.9818784981e-02
iterations=25000
Energy [2]:
Numeric=1.1202052285e-01
Analytic=1.1191465157e-01
iterations=16000
Energy [3]:
Numeric=1.9866346477e-01
Analytic=1.3006329597e-01
iterations=14000
Energy [4]:
Numeric=3.0934787926e-01
Analytic=1.9848619755e-01
iterations=11000
Energy [5]:
Numeric=4.4332771132e-01
Analytic=3.0908541971e-01
iterations=10000
Energy [6]:
Numeric=5.9921474507e-01
Analytic=4.4295950883e-01
iterations=9000
Energy [7]:
Numeric=7.7403857789e-01
Analytic=5.9874227213e-01
iterations=9000
Energy [8]:
Numeric=9.5707324477e-01
Analytic=7.7345405080e-01
iterations=13000
Energy [9]:
Numeric=1.0144712013e+00
Analytic=9.5619055627e-01
iterations=47000
Estamos em L = 51.01
Energy [0]:
Numeric=1.2164964307e-02
Analytic=1.2151247119e-02
iterations=35000
Energy [1]:
Numeric=4.8613622891e-02
Analytic=4.8562451255e-02
iterations=25000
Energy [2]:
Numeric=1.0919914568e-01
Analytic=1.0909919204e-01
iterations=17000
Energy [3]:
Numeric=1.9368668396e-01
Analytic=1.9351182174e-01
iterations=14000
Energy [4]:
Numeric=3.0164763414e-01
Analytic=3.0138425968e-01
iterations=11000
Energy [5]:
Numeric=4.3238381088e-01
Analytic=4.3202324362e-01
iterations=10000
Energy [6]:
Numeric=5.8465035929e-01
Analytic=5.8418923036e-01
iterations=9000
Energy [7]:
Numeric=7.5582802654e-01
Analytic=7.5525967045e-01
iterations=9000
Energy [8]:
Numeric=9.3741567436e-01
Analytic=9.3659421702e-01
iterations=11000
Estamos em L = 51.72
Energy [0]:
Numeric=1.1861516569e-02
Analytic=1.1848225685e-02
iterations=36000
Energy [1]:
Numeric=4.7401691819e-02
Analytic=4.7352993405e-02
iterations=26000
Energy [2]:
Numeric=1.0648819545e-01
Analytic=1.0638834954e-01
iterations=17000
Energy [3]:
Numeric=1.8888646785e-01
Analytic=1.8872102443e-01
iterations=15000
Energy [4]:
Numeric=2.9421077241e-01
Analytic=2.9396411327e-01
iterations=12000
Energy [5]:
Numeric=4.2182445013e-01
Analytic=4.2147813430e-01
iterations=11000
Energy [6]:
Numeric=5.7059800489e-01
Analytic=5.7013553074e-01
iterations=9000
Energy [7]:
Numeric=7.3818637601e-01
Analytic=7.3761579347e-01
iterations=9000
Energy [8]:
Numeric=9.1765994963e-01
Analytic=9.1689479954e-01
iterations=10000
Estamos em L = 52.42
Energy [0]:
Numeric=1.1569292863e-02
Analytic=1.1556396401e-02
iterations=37000
Energy [1]:
Numeric=4.6234681541e-02
Analytic=4.6188111470e-02
iterations=27000
Energy [2]:
Numeric=1.0387720712e-01
Analytic=1.0377702378e-01
iterations=17000
Energy [3]:
Numeric=1.8427051066e-01
Analytic=1.8410495625e-01
iterations=15000
Energy [4]:
Numeric=2.8705247446e-01
Analytic=2.8681171039e-01
iterations=12000
Energy [5]:
Numeric=4.1164473699e-01
Analytic=4.1130648095e-01
iterations=11000
Energy [6]:
Numeric=5.5701477558e-01
Analytic=5.5656108078e-01
iterations=9000
Energy [7]:
Numeric=7.2107063738e-01
Analytic=7.2051242290e-01
iterations=9000
Energy [8]:
Numeric=8.9803150966e-01
Analytic=8.9734169566e-01
iterations=10000
Estamos em L = 53.13
Energy [0]:
Numeric=1.1288556740e-02
Analytic=1.1275214972e-02
iterations=37000
Energy [1]:
Numeric=4.5112961043e-02
Analytic=4.5065644592e-02
iterations=27000
Energy [2]:
Numeric=1.0135412156e-01
Analytic=1.0126042007e-01
iterations=18000
Energy [3]:
Numeric=1.7981885331e-01
Analytic=1.7965528872e-01
iterations=15000
Energy [4]:
Numeric=2.8015603769e-01
Analytic=2.7991453380e-01
iterations=12000
Energy [5]:
Numeric=4.0182187266e-01
Analytic=4.0149149701e-01
iterations=11000
Energy [6]:
Numeric=5.4389184874e-01
Analytic=5.4344645549e-01
iterations=9000
Energy [7]:
Numeric=7.0448337257e-01
Analytic=7.0393722059e-01
iterations=9000
Energy [8]:
Numeric=8.7876143448e-01
Analytic=8.7807210253e-01
iterations=9000
Estamos em L = 53.84
Energy [0]:
Numeric=1.1017115364e-02
Analytic=1.1004169787e-02
iterations=38000
Energy [1]:
Numeric=4.4028799049e-02
Analytic=4.3983561060e-02
iterations=28000
Energy [2]:
Numeric=9.8927913078e-02
Analytic=9.8834027723e-02
iterations=18000
Energy [3]:
Numeric=1.7551911156e-01
Analytic=1.7536417843e-01
iterations=16000
Energy [4]:
Numeric=2.7349800850e-01
Analytic=2.7326077073e-01
iterations=12000
Energy [5]:
Numeric=3.9235016639e-01
Analytic=3.9201726326e-01
iterations=11000
Energy [6]:
Numeric=5.3119070850e-01
Analytic=5.3077293155e-01
iterations=10000
Energy [7]:
Numeric=6.8841069622e-01
Analytic=6.8787631323e-01
iterations=9000
Energy [8]:
Numeric=8.5983397394e-01
Analytic=8.5916578781e-01
iterations=9000
Estamos em L = 54.55
Energy [0]:
Numeric=1.0755358152e-02
Analytic=1.0742779590e-02
iterations=39000
Energy [1]:
Numeric=4.2986082175e-02
Analytic=4.2939949176e-02
iterations=28000
Energy [2]:
Numeric=9.6588093785e-02
Analytic=9.6493600231e-02
iterations=18000
Energy [3]:
Numeric=1.7137664463e-01
Analytic=1.7122423401e-01
iterations=16000
Energy [4]:
Numeric=2.6707260935e-01
Analytic=2.6683926779e-01
iterations=12000
Energy [5]:
Numeric=3.8319618003e-01
Analytic=3.8286868283e-01
iterations=11000
Energy [6]:
Numeric=5.1892960608e-01
Analytic=5.1852250418e-01
iterations=10000
Energy [7]:
Numeric=6.7283758065e-01
Analytic=6.7231486616e-01
iterations=9000
Energy [8]:
Numeric=8.4131152188e-01
Analytic=8.4066984642e-01
iterations=9000
Energy [9]:
Numeric=1.0031932322e+00
Analytic=9.9978391933e-01
iterations=31000
Estamos em L = 55.25
Energy [0]:
Numeric=1.0502828776e-02
Analytic=1.0490591333e-02
iterations=40000
Energy [1]:
Numeric=4.1977102802e-02
Analytic=4.1933008862e-02
iterations=29000
Energy [2]:
Numeric=9.4324166985e-02
Analytic=9.4235136999e-02
iterations=19000
Energy [3]:
Numeric=1.6738197242e-01
Analytic=1.6722848570e-01
iterations=16000
Energy [4]:
Numeric=2.6086199134e-01
Analytic=2.6063948898e-01
iterations=13000
Energy [5]:
Numeric=3.7433622144e-01
Analytic=3.7403143765e-01
iterations=12000
Energy [6]:
Numeric=5.0708809197e-01
Analytic=5.0667789137e-01
iterations=10000
Energy [7]:
Numeric=6.5776626920e-01
Analytic=6.5723749615e-01
iterations=9000
Energy [8]:
Numeric=8.2323249244e-01
Analytic=8.2261142535e-01
iterations=9000
Energy [9]:
Numeric=9.8919271889e-01
Analytic=9.8788806076e-01
iterations=20000
Estamos em L = 55.96
Energy [0]:
Numeric=1.0259097680e-02
Analytic=1.0247178216e-02
iterations=41000
Energy [1]:
Numeric=4.1003347803e-02
Analytic=4.0961043946e-02
iterations=30000
Energy [2]:
Numeric=9.2144302915e-02
Analytic=9.2054866554e-02
iterations=19000
Energy [3]:
Numeric=1.6351583560e-01
Analytic=1.6337035716e-01
iterations=17000
Energy [4]:
Numeric=2.5486957926e-01
Analytic=2.5465147655e-01
iterations=13000
Energy [5]:
Numeric=3.6579854992e-01
Analytic=3.6549194646e-01
iterations=12000
Energy [6]:
Numeric=4.9562450939e-01
Analytic=4.9522252872e-01
iterations=10000
Energy [7]:
Numeric=6.4312389358e-01
Analytic=6.4262857055e-01
iterations=10000
Energy [8]:
Numeric=8.0560844260e-01
Analytic=8.0500488198e-01
iterations=9000
Energy [9]:
Numeric=9.7246993834e-01
Analytic=9.7145189223e-01
iterations=15000
Estamos em L = 56.67
Energy [0]:
Numeric=1.0024521545e-02
Analytic=1.0012137912e-02
iterations=41000
Energy [1]:
Numeric=4.0065584918e-02
Analytic=4.0022455061e-02
iterations=30000
Energy [2]:
Numeric=9.0033136556e-02
Analytic=8.9949231047e-02
iterations=20000
Energy [3]:
Numeric=1.5978702413e-01
Analytic=1.0281106176e-01
iterations=17000
Energy [4]:
Numeric=2.4908528241e-01
Analytic=1.5964363948e-01
iterations=13000
Energy [5]:
Numeric=3.5753813940e-01
Analytic=2.4886581459e-01
iterations=12000
Energy [6]:
Numeric=4.8453468067e-01
Analytic=3.5723732466e-01
iterations=10000
Energy [7]:
Numeric=6.2895672009e-01
Analytic=4.8414055754e-01
iterations=10000
Energy [8]:
Numeric=7.8844392264e-01
Analytic=6.2847242482e-01
iterations=9000
Energy [9]:
Numeric=9.5471018262e-01
Analytic=7.8785608853e-01
iterations=13000
Energy [10]:
Numeric=1.0020270551e+00
Analytic=9.5385514478e-01
iterations=83000
Estamos em L = 57.37
Energy [0]:
Numeric=9.7971550923e-03
Analytic=9.7850908701e-03
iterations=42000
Energy [1]:
Numeric=3.9157095845e-02
Analytic=3.9115733095e-02
iterations=31000
Energy [2]:
Numeric=8.7999088944e-02
Analytic=8.7914871970e-02
iterations=20000
Energy [3]:
Numeric=1.5618696245e-01
Analytic=1.5604246708e-01
iterations=17000
Energy [4]:
Numeric=2.4349015688e-01
Analytic=2.4327359511e-01
iterations=13000
Energy [5]:
Numeric=3.4955070084e-01
Analytic=3.4925534594e-01
iterations=12000
Energy [6]:
Numeric=4.7380330614e-01
Analytic=4.7341680816e-01
iterations=10000
Energy [7]:
Numeric=6.1522680711e-01
Analytic=6.1475352053e-01
iterations=10000
Energy [8]:
Numeric=7.7173825995e-01
Analytic=7.7116515567e-01
iterations=9000
Energy [9]:
Numeric=9.3665585832e-01
Analytic=9.3589801321e-01
iterations=12000
Estamos em L = 58.08
Energy [0]:
Numeric=9.5774452768e-03
Analytic=9.5656788045e-03
iterations=43000
Energy [1]:
Numeric=3.8281792153e-02
Analytic=3.8239453166e-02
iterations=31000
Energy [2]:
Numeric=8.6033508148e-02
Analytic=8.5948616957e-02
iterations=20000
Energy [3]:
Numeric=1.5269574629e-01
Analytic=1.5256129545e-01
iterations=18000
Energy [4]:
Numeric=2.3806758437e-01
Analytic=2.3786638642e-01
iterations=14000
Energy [5]:
Numeric=3.4183362085e-01
Analytic=3.4153440579e-01
iterations=12000
Energy [6]:
Numeric=4.6342859001e-01
Analytic=4.6303677978e-01
iterations=10000
Energy [7]:
Numeric=6.0191893546e-01
Analytic=6.0145655972e-01
iterations=10000
Energy [8]:
Numeric=7.5548735169e-01
Analytic=7.5492822579e-01
iterations=9000
Energy [9]:
Numeric=9.1863588544e-01
Analytic=9.1791793682e-01
iterations=11000
Estamos em L = 58.79
Energy [0]:
Numeric=9.3650515556e-03
Analytic=9.3535632688e-03
iterations=44000
Energy [1]:
Numeric=3.7432856699e-02
Analytic=3.7392269044e-02
iterations=32000
Energy [2]:
Numeric=8.4127678837e-02
Analytic=8.4047467597e-02
iterations=21000
Energy [3]:
Numeric=1.4933014575e-01
Analytic=1.4919488048e-01
iterations=18000
Energy [4]:
Numeric=2.3283882365e-01
Analytic=2.3263620372e-01
iterations=14000
Energy [5]:
Numeric=3.3434145174e-01
Analytic=3.3406348689e-01
iterations=13000
Energy [6]:
Numeric=4.5335188612e-01
Analytic=4.5298661810e-01
iterations=11000
Energy [7]:
Numeric=5.8903591419e-01
Analytic=5.8856656656e-01
iterations=10000
Energy [8]:
Numeric=7.3970535110e-01
Analytic=7.3913869126e-01
iterations=9000
Energy [9]:
Numeric=9.0079672948e-01
Analytic=9.0008885656e-01
iterations=10000
Estamos em L = 59.49
Energy [0]:
Numeric=9.1596522209e-03
Analytic=9.1484243520e-03
iterations=45000
Energy [1]:
Numeric=3.6611936673e-02
Analytic=3.6572908011e-02
iterations=33000
Energy [2]:
Numeric=8.2289334475e-02
Analytic=8.2208588159e-02
iterations=21000
Energy [3]:
Numeric=1.4607473161e-01
Analytic=1.4593825940e-01
iterations=18000
Energy [4]:
Numeric=2.2777496175e-01
Analytic=2.2757548167e-01
iterations=14000
Energy [5]:
Numeric=3.2710440133e-01
Analytic=3.2683212633e-01
iterations=13000
Energy [6]:
Numeric=4.4361068844e-01
Analytic=4.4325309154e-01
iterations=11000
Energy [7]:
Numeric=5.7653034647e-01
Analytic=5.7606894473e-01
iterations=10000
Energy [8]:
Numeric=7.2434423931e-01
Analytic=7.2378804320e-01
iterations=9000
Energy [9]:
Numeric=8.8316885853e-01
Analytic=8.8250967985e-01
iterations=10000
Estamos em L = 60.20
Energy [0]:
Numeric=8.9609431456e-03
Analytic=8.9499595177e-03
iterations=46000
Energy [1]:
Numeric=3.5820139720e-02
Analytic=3.5780166099e-02
iterations=33000
Energy [2]:
Numeric=8.0510934009e-02
Analytic=8.0429295158e-02
iterations=21000
Energy [3]:
Numeric=1.4291375719e-01
Analytic=1.4278673297e-01
iterations=19000
Energy [4]:
Numeric=2.2287367666e-01
Analytic=2.2267704883e-01
iterations=14000
Energy [5]:
Numeric=3.2010587075e-01
Analytic=3.1983038466e-01
iterations=13000
Energy [6]:
Numeric=4.3417367839e-01
Analytic=4.3382356665e-01
iterations=11000
Energy [7]:
Numeric=5.6440338647e-01
Analytic=5.6394951640e-01
iterations=10000
Energy [8]:
Numeric=7.0941259158e-01
Analytic=7.0886647458e-01
iterations=9000
Energy [9]:
Numeric=8.6586491698e-01
Analytic=8.6523941220e-01
iterations=10000
Estamos em L = 60.91
Energy [0]:
Numeric=8.7686366284e-03
Analytic=8.7578823942e-03
iterations=47000
Energy [1]:
Numeric=3.5051324811e-02
Analytic=3.5012903696e-02
iterations=34000
Energy [2]:
Numeric=7.8784046842e-02
Analytic=7.8707047684e-02
iterations=22000
Energy [3]:
Numeric=1.3986382164e-01
Analytic=1.3973584903e-01
iterations=19000
Energy [4]:
Numeric=2.1812171373e-01
Analytic=2.1793410385e-01
iterations=15000
Energy [5]:
Numeric=3.1332055574e-01
Analytic=3.1304881657e-01
iterations=13000
Energy [6]:
Numeric=4.2504101511e-01
Analytic=4.2468598316e-01
iterations=11000
Energy [7]:
Numeric=5.5261650417e-01
Analytic=5.5219454739e-01
iterations=11000
Energy [8]:
Numeric=6.9489984965e-01
Analytic=6.9436331563e-01
iterations=9000
Energy [9]:
Numeric=8.4893762193e-01
Analytic=8.4831373553e-01
iterations=10000
Energy [10]:
Numeric=9.9930815763e-01
Analytic=9.9717234234e-01
iterations=25000
Estamos em L = 61.62
Energy [0]:
Numeric=8.5831408514e-03
Analytic=8.5719218480e-03
iterations=47000
Energy [1]:
Numeric=3.4309558034e-02
Analytic=3.4270041453e-02
iterations=34000
Energy [2]:
Numeric=7.7116536426e-02
Analytic=7.7039438445e-02
iterations=22000
Energy [3]:
Numeric=1.3690823510e-01
Analytic=1.3678138724e-01
iterations=19000
Energy [4]:
Numeric=2.1352451247e-01
Analytic=2.1334019326e-01
iterations=15000
Energy [5]:
Numeric=3.0673053409e-01
Analytic=3.0647844338e-01
iterations=14000
Energy [6]:
Numeric=4.1617867132e-01
Analytic=4.1582882921e-01
iterations=11000
Energy [7]:
Numeric=5.4121838093e-01
Analytic=5.4079076172e-01
iterations=11000
Energy [8]:
Numeric=6.8076625278e-01
Analytic=6.8026735196e-01
iterations=10000
Energy [9]:
Numeric=8.3235949358e-01
Analytic=8.3175378638e-01
iterations=10000
Energy [10]:
Numeric=9.8503187998e-01
Analytic=9.8386767731e-01
iterations=18000
Estamos em L = 62.32
Energy [0]:
Numeric=8.4028107841e-03
Analytic=8.3918209413e-03
iterations=48000
Energy [1]:
Numeric=3.3588521671e-02
Analytic=3.3550556520e-02
iterations=35000
Energy [2]:
Numeric=7.5497133165e-02
Analytic=3.7859515977e-02
iterations=23000
Energy [3]:
Numeric=1.3403966672e-01
Analytic=7.5424185450e-02
iterations=20000
Energy [4]:
Numeric=2.0907565235e-01
Analytic=1.3391934482e-01
iterations=15000
Energy [5]:
Numeric=3.0036572943e-01
Analytic=2.0888919071e-01
iterations=14000
Energy [6]:
Numeric=4.0756707995e-01
Analytic=3.0011072695e-01
iterations=12000
Energy [7]:
Numeric=5.3014512305e-01
Analytic=4.0724111673e-01
iterations=11000
Energy [8]:
Numeric=6.6707291553e-01
Analytic=5.2972534830e-01
iterations=10000
Energy [9]:
Numeric=8.1616108290e-01
Analytic=6.6656705912e-01
iterations=10000
Energy [10]:
Numeric=9.6918917905e-01
Analytic=8.1557120455e-01
iterations=14000
Energy [11]:
Numeric=1.0103047954e+00
Analytic=9.6822712026e-01
iterations=58000
Estamos em L = 63.03
Energy [0]:
Numeric=8.2281110283e-03
Analytic=8.2173361118e-03
iterations=49000
Energy [1]:
Numeric=3.2890049052e-02
Analytic=3.2853479027e-02
iterations=36000
Energy [2]:
Numeric=7.3932760664e-02
Analytic=7.3859124294e-02
iterations=23000
Energy [3]:
Numeric=1.3126735512e-01
Analytic=1.3114592339e-01
iterations=20000
Energy [4]:
Numeric=2.0475969631e-01
Analytic=2.0457527771e-01
iterations=15000
Energy [5]:
Numeric=2.9418845552e-01
Analytic=2.9393754527e-01
iterations=14000
Energy [6]:
Numeric=3.9923112392e-01
Analytic=3.9891235751e-01
iterations=12000
Energy [7]:
Numeric=5.1939806622e-01
Analytic=5.1898596153e-01
iterations=11000
Energy [8]:
Numeric=6.5374725521e-01
Analytic=6.5325077642e-01
iterations=10000
Energy [9]:
Numeric=8.0034663056e-01
Analytic=7.9977122264e-01
iterations=10000
Energy [10]:
Numeric=9.5272936639e-01
Analytic=9.5187349981e-01
iterations=12000
Estamos em L = 63.74
Energy [0]:
Numeric=8.0588094328e-03
Analytic=8.0482362994e-03
iterations=50000
Energy [1]:
Numeric=3.2215546749e-02
Analytic=3.2177888836e-02
iterations=36000
Energy [2]:
Numeric=7.2416844106e-02
Analytic=7.2342200982e-02
iterations=23000
Energy [3]:
Numeric=1.2857812240e-01
Analytic=1.2845751659e-01
iterations=20000
Energy [4]:
Numeric=2.0056361198e-01
Analytic=2.0039292547e-01
iterations=16000
Energy [5]:
Numeric=2.8819822814e-01
Analytic=2.8795116936e-01
iterations=14000
Energy [6]:
Numeric=3.9115607847e-01
Analytic=3.9083253978e-01
iterations=12000
Energy [7]:
Numeric=5.0896537278e-01
Analytic=5.0856071750e-01
iterations=11000
Energy [8]:
Numeric=6.4079423068e-01
Analytic=6.4030683627e-01
iterations=10000
Energy [9]:
Numeric=7.8491633294e-01
Analytic=7.8435464969e-01
iterations=10000
Energy [10]:
Numeric=9.3608919136e-01
Analytic=9.3531233492e-01
iterations=11000
Estamos em L = 64.44
Energy [0]:
Numeric=7.8946857343e-03
Analytic=7.8843022249e-03
iterations=51000
Energy [1]:
Numeric=3.1559170813e-02
Analytic=3.1522912511e-02
iterations=37000
Energy [2]:
Numeric=7.0941945170e-02
Analytic=7.0871465259e-02
iterations=24000
Energy [3]:
Numeric=1.2596498494e-01
Analytic=1.2585069873e-01
iterations=21000
Energy [4]:
Numeric=1.9650980156e-01
Analytic=1.9633687813e-01
iterations=16000
Energy [5]:
Numeric=2.8238114488e-01
Analytic=2.8214424157e-01
iterations=15000
Energy [6]:
Numeric=3.8331049101e-01
Analytic=3.8299210565e-01
iterations=12000
Energy [7]:
Numeric=4.9885054300e-01
Analytic=4.9843818663e-01
iterations=11000
Energy [8]:
Numeric=6.2820202146e-01
Analytic=6.2772366028e-01
iterations=10000
Energy [9]:
Numeric=7.6986772896e-01
Analytic=7.6931919411e-01
iterations=10000
Energy [10]:
Numeric=9.1946257493e-01
Analytic=9.1877527813e-01
iterations=11000
Estamos em L = 65.15
Energy [0]:
Numeric=7.7355308288e-03
Analytic=7.7253256186e-03
iterations=52000
Energy [1]:
Numeric=3.0922710865e-02
Analytic=3.0887720511e-02
iterations=38000
Energy [2]:
Numeric=6.9516449755e-02
Analytic=6.9445064407e-02
iterations=24000
Energy [3]:
Numeric=1.2343778869e-01
Analytic=1.2332221401e-01
iterations=21000
Energy [4]:
Numeric=1.9257280572e-01
Analytic=1.9240213696e-01
iterations=16000
Energy [5]:
Numeric=2.7674238365e-01
Analytic=2.7650975520e-01
iterations=15000
Energy [6]:
Numeric=3.7569538438e-01
Analytic=3.7538192928e-01
iterations=12000
Energy [7]:
Numeric=4.8899151337e-01
Analytic=4.8860738407e-01
iterations=12000
Energy [8]:
Numeric=6.1595931609e-01
Analytic=6.1548983035e-01
iterations=10000
Energy [9]:
Numeric=7.5521908691e-01
Analytic=7.5466037318e-01
iterations=10000
Energy [10]:
Numeric=9.0307099889e-01
Analytic=9.0238712956e-01
iterations=10000
Estamos em L = 65.86
Energy [0]:
Numeric=7.5811460944e-03
Analytic=7.5711086548e-03
iterations=53000
Energy [1]:
Numeric=3.0307613317e-02
Analytic=3.0271524564e-02
iterations=38000
Energy [2]:
Numeric=6.8128823641e-02
Analytic=6.8061237468e-02
iterations=25000
Energy [3]:
Numeric=1.2098398851e-01
Analytic=1.2086896665e-01
iterations=21000
Energy [4]:
Numeric=1.8875751830e-01
Analytic=1.8858394563e-01
iterations=16000
Energy [5]:
Numeric=2.7126953597e-01
Analytic=2.7104103528e-01
iterations=15000
Energy [6]:
Numeric=3.6830220503e-01
Analytic=3.6799329600e-01
iterations=12000
Energy [7]:
Numeric=4.7943437125e-01
Analytic=4.7905775808e-01
iterations=12000
Energy [8]:
Numeric=6.0405490723e-01
Analytic=6.0359414094e-01
iterations=10000
Energy [9]:
Numeric=7.4092107049e-01
Analytic=7.4037215376e-01
iterations=10000
Energy [10]:
Numeric=8.8686101995e-01
Analytic=8.8622096638e-01
iterations=10000
Estamos em L = 66.57
Energy [0]:
Numeric=7.4313427632e-03
Analytic=7.4214632412e-03
iterations=54000
Energy [1]:
Numeric=2.9708387628e-02
Analytic=2.9673575235e-02
iterations=39000
Energy [2]:
Numeric=6.6786725397e-02
Analytic=6.6718309855e-02
iterations=25000
Energy [3]:
Numeric=1.1859687443e-01
Analytic=1.1848801157e-01
iterations=22000
Energy [4]:
Numeric=1.8503842918e-01
Analytic=1.8487777646e-01
iterations=17000
Energy [5]:
Numeric=2.6596470009e-01
Analytic=2.6573172052e-01
iterations=15000
Energy [6]:
Numeric=3.6111394006e-01
Analytic=3.6081788226e-01
iterations=13000
Energy [7]:
Numeric=4.7014849810e-01
Analytic=4.6977917737e-01
iterations=12000
Energy [8]:
Numeric=5.9249785884e-01
Analytic=5.9202563673e-01
iterations=10000
Energy [9]:
Numeric=7.2698682385e-01
Analytic=7.2644741288e-01
iterations=10000
Energy [10]:
Numeric=8.7093066216e-01
Analytic=8.7032139801e-01
iterations=10000
Estamos em L = 67.27
Energy [0]:
Numeric=7.2859413371e-03
Analytic=7.2762104514e-03
iterations=55000
Energy [1]:
Numeric=2.9126808656e-02
Analytic=2.9093159661e-02
iterations=40000
Energy [2]:
Numeric=6.5479609846e-02
Analytic=6.5414688336e-02
iterations=26000
Energy [3]:
Numeric=1.1628688657e-01
Analytic=1.1617654574e-01
iterations=22000
Energy [4]:
Numeric=1.8144210531e-01
Analytic=1.8127931750e-01
iterations=17000
Energy [5]:
Numeric=2.6079202503e-01
Analytic=2.6057574631e-01
iterations=16000
Energy [6]:
Numeric=3.5413873366e-01
Analytic=3.5384773646e-01
iterations=13000
Energy [7]:
Numeric=4.6113811008e-01
Analytic=4.6076191743e-01
iterations=12000
Energy [8]:
Numeric=5.8121283955e-01
Analytic=5.8077363932e-01
iterations=11000
Energy [9]:
Numeric=7.1340857639e-01
Analytic=7.1287827407e-01
iterations=10000
Energy [10]:
Numeric=8.5532779426e-01
Analytic=8.5471597606e-01
iterations=10000
Energy [11]:
Numeric=9.9528651477e-01
Analytic=9.9367085627e-01
iterations=22000
Estamos em L = 67.98
Energy [0]:
Numeric=7.1447710440e-03
Analytic=7.1351800886e-03
iterations=56000
Energy [1]:
Numeric=2.8564370809e-02
Analytic=2.8529599416e-02
iterations=40000
Energy [2]:
Numeric=6.4214550689e-02
Analytic=6.4148856344e-02
iterations=26000
Energy [3]:
Numeric=1.1404194022e-01
Analytic=1.1393190008e-01
iterations=22000
Energy [4]:
Numeric=1.7794566289e-01
Analytic=1.7778446051e-01
iterations=17000
Energy [5]:
Numeric=2.5577931957e-01
Analytic=2.5556732877e-01
iterations=16000
Energy [6]:
Numeric=3.4736133798e-01
Analytic=3.4707526064e-01
iterations=13000
Energy [7]:
Numeric=4.5236783699e-01
Analytic=4.5199664648e-01
iterations=12000
Energy [8]:
Numeric=5.7025842237e-01
Analytic=5.6982776526e-01
iterations=11000
Energy [9]:
Numeric=7.0017764866e-01
Analytic=6.9965635609e-01
iterations=10000
Energy [10]:
Numeric=8.4001606887e-01
Analytic=8.3942140475e-01
iterations=10000
Energy [11]:
Numeric=9.8125813956e-01
Analytic=9.8022271434e-01
iterations=17000
Estamos em L = 68.69
Energy [0]:
Numeric=7.0076693310e-03
Analytic=6.9982100919e-03
iterations=57000
Energy [1]:
Numeric=2.8015845268e-02
Analytic=2.7982248553e-02
iterations=41000
Energy [2]:
Numeric=6.2986106101e-02
Analytic=6.2919369611e-02
iterations=26000
Energy [3]:
Numeric=1.1185553409e-01
Analytic=1.1175153189e-01
iterations=23000
Energy [4]:
Numeric=1.7454915460e-01
Analytic=1.7438928960e-01
iterations=17000
Energy [5]:
Numeric=2.5091722742e-01
Analytic=2.5070094967e-01
iterations=16000
Energy [6]:
Numeric=3.4077455143e-01
Analytic=3.4049319314e-01
iterations=13000
Energy [7]:
Numeric=4.4384102514e-01
Analytic=4.4347441113e-01
iterations=12000
Energy [8]:
Numeric=5.5960015499e-01
Analytic=5.5917793753e-01
iterations=11000
Energy [9]:
Numeric=6.8725503004e-01
Analytic=6.8677295883e-01
iterations=11000
Energy [10]:
Numeric=8.2502706438e-01
Analytic=8.2444718769e-01
iterations=10000
Energy [11]:
Numeric=9.6631814407e-01
Analytic=9.6542132955e-01
iterations=14000
Estamos em L = 69.39
Energy [0]:
Numeric=6.8744813930e-03
Analytic=6.8651461024e-03
iterations=58000
Energy [1]:
Numeric=2.7483012248e-02
Analytic=2.7450491746e-02
iterations=42000
Energy [2]:
Numeric=6.1788042394e-02
Analytic=6.1724852075e-02
iterations=27000
Energy [3]:
Numeric=1.0973869430e-01
Analytic=1.0963301777e-01
iterations=23000
Energy [4]:
Numeric=1.7124177057e-01
Analytic=1.7109007070e-01
iterations=18000
Energy [5]:
Numeric=2.4618481504e-01
Analytic=2.4597134236e-01
iterations=16000
Energy [6]:
Numeric=3.3438317896e-01
Analytic=3.3409459194e-01
iterations=13000
Energy [7]:
Numeric=4.3552507419e-01
Analytic=4.3518662193e-01
iterations=13000
Energy [8]:
Numeric=5.4922848540e-01
Analytic=5.4881439193e-01
iterations=11000
Energy [9]:
Numeric=6.7471324586e-01
Analytic=6.7421920309e-01
iterations=11000
Energy [10]:
Numeric=8.1036440382e-01
Analytic=8.0979789675e-01
iterations=10000
Energy [11]:
Numeric=9.5093787545e-01
Analytic=9.5017323257e-01
iterations=13000
Estamos em L = 70.10
Energy [0]:
Numeric=6.7450597318e-03
Analytic=6.7358410030e-03
iterations=59000
Energy [1]:
Numeric=2.6965272489e-02
Analytic=2.6933742572e-02
iterations=43000
Energy [2]:
Numeric=6.0628164737e-02
Analytic=6.0563992061e-02
iterations=27000
Energy [3]:
Numeric=1.0767429222e-01
Analytic=1.0757404704e-01
iterations=24000
Energy [4]:
Numeric=1.6803319826e-01
Analytic=1.6788324162e-01
iterations=18000
Energy [5]:
Numeric=2.4157800522e-01
Analytic=2.4137347839e-01
iterations=17000
Energy [6]:
Numeric=3.2813949062e-01
Analytic=2.6999663018e-01
iterations=14000
Energy [7]:
Numeric=4.2747011294e-01
Analytic=3.2787281897e-01
iterations=13000
Energy [8]:
Numeric=5.3915243328e-01
Analytic=4.2712503906e-01
iterations=11000
Energy [9]:
Numeric=6.6247154426e-01
Analytic=5.3872767941e-01
iterations=11000
Energy [10]:
Numeric=7.9602893128e-01
Analytic=6.6198613640e-01
iterations=10000
Energy [11]:
Numeric=9.3552250211e-01
Analytic=7.9547464888e-01
iterations=12000
Energy [12]:
Numeric=1.0027419194e+00
Analytic=9.3481291053e-01
iterations=102000
Estamos em L = 70.81
Energy [0]:
Numeric=6.6192637446e-03
Analytic=6.6101545936e-03
iterations=60000
Energy [1]:
Numeric=2.6464029504e-02
Analytic=2.6431441896e-02
iterations=43000
Energy [2]:
Numeric=5.9496416666e-02
Analytic=5.9435538712e-02
iterations=28000
Energy [3]:
Numeric=1.0567207493e-01
Analytic=1.0557241546e-01
iterations=24000
Energy [4]:
Numeric=1.6491826363e-01
Analytic=1.6476540270e-01
iterations=18000
Energy [5]:
Numeric=2.3710385128e-01
Analytic=2.3690255503e-01
iterations=17000
Energy [6]:
Numeric=3.2208343163e-01
Analytic=3.2182152512e-01
iterations=14000
Energy [7]:
Numeric=4.1962173606e-01
Analytic=4.1928175820e-01
iterations=13000
Energy [8]:
Numeric=5.2932769624e-01
Analytic=5.2890866549e-01
iterations=11000
Energy [9]:
Numeric=6.5054158511e-01
Analytic=6.5006481304e-01
iterations=11000
Energy [10]:
Numeric=7.8201846285e-01
Analytic=7.8147610287e-01
iterations=10000
Energy [11]:
Numeric=9.2019691463e-01
Analytic=9.1950425697e-01
iterations=11000
Estamos em L = 71.52
Energy [0]:
Numeric=6.4975128646e-03
Analytic=6.4879530696e-03
iterations=60000
Energy [1]:
Numeric=2.5974641338e-02
Analytic=2.5943056363e-02
iterations=44000
Energy [2]:
Numeric=5.8399578439e-02
Analytic=5.8338298639e-02
iterations=28000
Energy [3]:
Numeric=1.0372755405e-01
Analytic=1.0362601953e-01
iterations=24000
Energy [4]:
Numeric=1.6187478067e-01
Analytic=1.6173330816e-01
iterations=19000
Energy [5]:
Numeric=2.3275222367e-01
Analytic=2.3255398347e-01
iterations=17000
Energy [6]:
Numeric=3.1620228028e-01
Analytic=3.1593463603e-01
iterations=14000
Energy [7]:
Numeric=4.1198432540e-01
Analytic=4.1164919669e-01
iterations=13000
Energy [8]:
Numeric=5.1976230299e-01
Analytic=5.1934852724e-01
iterations=11000
Energy [9]:
Numeric=6.3891458950e-01
Analytic=6.3844635480e-01
iterations=11000
Energy [10]:
Numeric=7.6832992321e-01
Analytic=7.6779915199e-01
iterations=10000
Energy [11]:
Numeric=9.0497849305e-01
Analytic=9.0433900065e-01
iterations=11000
Estamos em L = 72.22
Energy [0]:
Numeric=6.3785626091e-03
Analytic=6.3691088436e-03
iterations=61000
Energy [1]:
Numeric=2.5498735636e-02
Analytic=2.5468076984e-02
iterations=45000
Energy [2]:
Numeric=5.7329340419e-02
Analytic=5.7271132791e-02
iterations=29000
Energy [3]:
Numeric=1.0182900450e-01
Analytic=1.0173285095e-01
iterations=25000
Energy [4]:
Numeric=1.5892765765e-01
Analytic=1.5878385783e-01
iterations=19000
Energy [5]:
Numeric=2.2852626067e-01
Analytic=2.2832337773e-01
iterations=17000
Energy [6]:
Numeric=3.1047082505e-01
Analytic=3.1020633860e-01
iterations=14000
Energy [7]:
Numeric=4.0455069941e-01
Analytic=4.0422008005e-01
iterations=13000
Energy [8]:
Numeric=5.1042038296e-01
Analytic=5.1003874852e-01
iterations=12000
Energy [9]:
Numeric=6.2758178779e-01
Analytic=6.2712199675e-01
iterations=11000
Energy [10]:
Numeric=7.5498539972e-01
Analytic=7.5443941701e-01
iterations=10000
Energy [11]:
Numeric=8.8997617973e-01
Analytic=8.8937227410e-01
iterations=11000
Estamos em L = 72.93
Energy [0]:
Numeric=6.2628546832e-03
Analytic=6.2535000222e-03
iterations=62000
Energy [1]:
Numeric=2.5037784364e-02
Analytic=2.5006017823e-02
iterations=45000
Energy [2]:
Numeric=5.6292075218e-02
Analytic=5.6232953515e-02
iterations=29000
Energy [3]:
Numeric=9.9986786199e-02
Analytic=9.9890991570e-02
iterations=25000
Energy [4]:
Numeric=1.5605670778e-01
Analytic=1.5591408947e-01
iterations=19000
Energy [5]:
Numeric=2.2439422036e-01
Analytic=2.2420654417e-01
iterations=18000
Energy [6]:
Numeric=3.0489259533e-01
Analytic=3.0463106825e-01
iterations=14000
Energy [7]:
Numeric=3.9730504777e-01
Analytic=3.9698742885e-01
iterations=14000
Energy [8]:
Numeric=5.0134487408e-01
Analytic=5.0097111393e-01
iterations=12000
Energy [9]:
Numeric=6.1655642356e-01
Analytic=6.1608312155e-01
iterations=11000
Energy [10]:
Numeric=7.4192937529e-01
Analytic=7.4139160400e-01
iterations=10000
Energy [11]:
Numeric=8.7521644991e-01
Analytic=8.7463843335e-01
iterations=11000
Energy [12]:
Numeric=1.0032689427e+00
Analytic=9.9986266841e-01
iterations=28000
Estamos em L = 73.64
Energy [0]:
Numeric=6.1502723960e-03
Analytic=6.1410102444e-03
iterations=63000
Energy [1]:
Numeric=2.4587241385e-02
Analytic=2.4556414736e-02
iterations=46000
Energy [2]:
Numeric=5.5282993440e-02
Analytic=5.5222721811e-02
iterations=29000
Energy [3]:
Numeric=9.8189635514e-02
Analytic=9.8098608584e-02
iterations=26000
Energy [4]:
Numeric=1.5326281343e-01
Analytic=1.5312117156e-01
iterations=19000
Energy [5]:
Numeric=2.2039069100e-01
Analytic=2.2019947167e-01
iterations=18000
Energy [6]:
Numeric=2.9944367741e-01
Analytic=2.9920349676e-01
iterations=15000
Energy [7]:
Numeric=3.9025715055e-01
Analytic=3.8994454604e-01
iterations=14000
Energy [8]:
Numeric=4.9252203910e-01
Analytic=4.9213770166e-01
iterations=12000
Energy [9]:
Numeric=6.0578909845e-01
Analytic=6.0532128489e-01
iterations=11000
Energy [10]:
Numeric=7.2917989537e-01
Analytic=7.2864976825e-01
iterations=10000
Energy [11]:
Numeric=8.6071610290e-01
Analytic=8.6015909890e-01
iterations=11000
Energy [12]:
Numeric=9.9147236927e-01
Analytic=9.9010191133e-01
iterations=19000
Estamos em L = 74.34
Energy [0]:
Numeric=6.0407042690e-03
Analytic=6.0315283633e-03
iterations=64000
Energy [1]:
Numeric=2.4148779746e-02
Analytic=2.4118824235e-02
iterations=47000
Energy [2]:
Numeric=5.4296567938e-02
Analytic=5.4239444740e-02
iterations=30000
Energy [3]:
Numeric=9.6446467737e-02
Analytic=9.6353950034e-02
iterations=26000
Energy [4]:
Numeric=1.5053688888e-01
Analytic=1.5040239647e-01
iterations=20000
Energy [5]:
Numeric=2.1648735144e-01
Analytic=2.1629832234e-01
iterations=18000
Energy [6]:
Numeric=2.9416481628e-01
Analytic=2.9391852080e-01
iterations=15000
Energy [7]:
Numeric=3.8339271235e-01
Analytic=3.8308500473e-01
iterations=14000
Energy [8]:
Numeric=4.8390978249e-01
Analytic=4.8353087576e-01
iterations=12000
Energy [9]:
Numeric=5.9526043231e-01
Analytic=5.9482823390e-01
iterations=12000
Energy [10]:
Numeric=7.1669771329e-01
Analytic=7.1620751149e-01
iterations=11000
Energy [11]:
Numeric=8.4651653972e-01
Analytic=8.4594764557e-01
iterations=11000
Energy [12]:
Numeric=9.7798345432e-01
Analytic=9.7695082777e-01
iterations=15000
Estamos em L = 75.05
Energy [0]:
Numeric=5.9340437582e-03
Analytic=5.9249480708e-03
iterations=65000
Energy [1]:
Numeric=2.3721967793e-02
Analytic=2.3692822380e-02
iterations=48000
Energy [2]:
Numeric=5.3340406589e-02
Analytic=5.3282173008e-02
iterations=30000
Energy [3]:
Numeric=9.4749661763e-02
Analytic=9.4655340585e-02
iterations=26000
Energy [4]:
Numeric=1.4788835043e-01
Analytic=1.4775517407e-01
iterations=20000
Energy [5]:
Numeric=2.1268645853e-01
Analytic=2.1249942274e-01
iterations=18000
Energy [6]:
Numeric=2.8901427040e-01
Analytic=2.8877125103e-01
iterations=15000
Energy [7]:
Numeric=3.7670561841e-01
Analytic=3.7640263638e-01
iterations=14000
Energy [8]:
Numeric=4.7551685217e-01
Analytic=4.7514327781e-01
iterations=12000
Energy [9]:
Numeric=5.8501995242e-01
Analytic=5.8459592017e-01
iterations=12000
Energy [10]:
Numeric=7.0453838808e-01
Analytic=7.0405813083e-01
iterations=11000
Energy [11]:
Numeric=8.3256752636e-01
Analytic=8.3201189078e-01
iterations=11000
Energy [12]:
Numeric=9.6383609179e-01
Analytic=9.6296595183e-01
iterations=13000
Estamos em L = 75.76
Energy [0]:
Numeric=5.8296815137e-03
Analytic=5.8211677395e-03
iterations=67000
Energy [1]:
Numeric=2.3308236648e-02
Analytic=2.3278003751e-02
iterations=48000
Energy [2]:
Numeric=5.2405757668e-02
Analytic=5.2349998680e-02
iterations=31000
Energy [3]:
Numeric=9.3088773801e-02
Analytic=9.3001177553e-02
iterations=27000
Energy [4]:
Numeric=1.4531328628e-01
Analytic=1.4517702572e-01
iterations=20000
Energy [5]:
Numeric=2.0897804291e-01
Analytic=2.0879925568e-01
iterations=19000
Energy [6]:
Numeric=2.8399703251e-01
Analytic=2.8375700181e-01
iterations=15000
Energy [7]:
Numeric=3.7020323951e-01
Analytic=3.6989151951e-01
iterations=14000
Energy [8]:
Numeric=4.6733628947e-01
Analytic=4.6696781834e-01
iterations=12000
Energy [9]:
Numeric=5.7503261930e-01
Analytic=5.7461650844e-01
iterations=12000
Energy [10]:
Numeric=6.9266501314e-01
Analytic=6.9219473212e-01
iterations=11000
Energy [11]:
Numeric=8.1889926163e-01
Analytic=8.1835579327e-01
iterations=11000
Energy [12]:
Numeric=9.4946558755e-01
Analytic=9.4869977365e-01
iterations=12000
Estamos em L = 76.46
Energy [0]:
Numeric=5.7285405632e-03
Analytic=5.7200901620e-03
iterations=68000
Energy [1]:
Numeric=2.2903390023e-02
Analytic=2.2873980497e-02
iterations=49000
Energy [2]:
Numeric=5.1498884381e-02
Analytic=5.1442053047e-02
iterations=31000
Energy [3]:
Numeric=9.1479231123e-02
Analytic=9.1389927172e-02
iterations=27000
Energy [4]:
Numeric=1.4279141355e-01
Analytic=1.4266557861e-01
iterations=21000
Energy [5]:
Numeric=2.0537085674e-01
Analytic=2.0519445238e-01
iterations=19000
Energy [6]:
Numeric=2.7910171835e-01
Analytic=2.7887128140e-01
iterations=16000
Energy [7]:
Numeric=3.6383411156e-01
Analytic=3.6354596881e-01
iterations=15000
Energy [8]:
Numeric=4.5935242954e-01
Analytic=4.5899766803e-01
iterations=13000
Energy [9]:
Numeric=5.6531109009e-01
Analytic=5.6488238195e-01
iterations=12000
Energy [10]:
Numeric=6.8109575442e-01
Analytic=6.8061031678e-01
iterations=11000
Energy [11]:
Numeric=8.0551278812e-01
Analytic=8.0498057321e-01
iterations=11000
Energy [12]:
Numeric=9.3504540574e-01
Analytic=9.3438206241e-01
iterations=12000
Estamos em L = 77.17
Energy [0]:
Numeric=5.6300141810e-03
Analytic=5.6216223486e-03
iterations=69000
Energy [1]:
Numeric=2.2509023471e-02
Analytic=2.2480381430e-02
iterations=50000
Energy [2]:
Numeric=5.0611528211e-02
Analytic=5.0557504611e-02
iterations=32000
Energy [3]:
Numeric=8.9905011284e-02
Analytic=8.9820121083e-02
iterations=28000
Energy [4]:
Numeric=1.4034699186e-01
Analytic=1.4021856039e-01
iterations=21000
Energy [5]:
Numeric=2.0186230159e-01
Analytic=2.0168178516e-01
iterations=19000
Energy [6]:
Numeric=2.7433685877e-01
Analytic=2.7410978271e-01
iterations=16000
Energy [7]:
Numeric=3.5764386519e-01
Analytic=3.5736052474e-01
iterations=15000
Energy [8]:
Numeric=4.5157548918e-01
Analytic=4.5122624886e-01
iterations=13000
Energy [9]:
Numeric=5.5580933453e-01
Analytic=5.5538614511e-01
iterations=12000
Energy [10]:
Numeric=6.6977599047e-01
Analytic=6.6929784859e-01
iterations=11000
Energy [11]:
Numeric=7.9240662346e-01
Analytic=7.9188547664e-01
iterations=11000
Energy [12]:
Numeric=9.2078679737e-01
Analytic=9.2013179137e-01
iterations=11000
Estamos em L = 77.88
Energy [0]:
Numeric=5.5340131470e-03
Analytic=5.5256751446e-03
iterations=70000
Energy [1]:
Numeric=2.2124775860e-02
Analytic=2.2096851169e-02
iterations=51000
Energy [2]:
Numeric=4.9750631569e-02
Analytic=4.9695557190e-02
iterations=32000
Energy [3]:
Numeric=8.8376759811e-02
Analytic=8.8290353024e-02
iterations=28000
Energy [4]:
Numeric=1.3796139774e-01
Analytic=1.3783379417e-01
iterations=21000
Energy [5]:
Numeric=1.9843741359e-01
Analytic=1.9825816048e-01
iterations=19000
Energy [6]:
Numeric=2.6969216678e-01
Analytic=2.6946837453e-01
iterations=16000
Energy [7]:
Numeric=3.5160852741e-01
Analytic=3.5132994358e-01
iterations=15000
Energy [8]:
Numeric=4.4399110966e-01
Analytic=4.4364722527e-01
iterations=13000
Energy [9]:
Numeric=5.4653854378e-01
Analytic=5.4612062415e-01
iterations=12000
Energy [10]:
Numeric=6.5872131427e-01
Analytic=6.5825030506e-01
iterations=11000
Energy [11]:
Numeric=7.7957862551e-01
Analytic=7.7906831209e-01
iterations=11000
Energy [12]:
Numeric=9.0662609557e-01
Analytic=9.0601779752e-01
iterations=11000
Estamos em L = 78.59
Energy [0]:
Numeric=5.4404520194e-03
Analytic=5.4321633058e-03
iterations=71000
Energy [1]:
Numeric=2.1752061138e-02
Analytic=2.1723049337e-02
iterations=51000
Energy [2]:
Numeric=4.8907856278e-02
Analytic=4.8855448139e-02
iterations=33000
Energy [3]:
Numeric=8.6885758187e-02
Analytic=8.6799275718e-02
iterations=28000
Energy [4]:
Numeric=1.3563074847e-01
Analytic=1.3550919379e-01
iterations=22000
Energy [5]:
Numeric=1.9508538888e-01
Analytic=1.9492061242e-01
iterations=20000
Energy [6]:
Numeric=2.6517289133e-01
Analytic=2.6494309318e-01
iterations=16000
Energy [7]:
Numeric=3.4573636370e-01
Analytic=3.4544918786e-01
iterations=15000
Energy [8]:
Numeric=4.3659336722e-01
Analytic=4.3625449538e-01
iterations=13000
Energy [9]:
Numeric=5.3746451599e-01
Analytic=5.3707886602e-01
iterations=13000
Energy [10]:
Numeric=6.4792475850e-01
Analytic=6.4746071705e-01
iterations=11000
Energy [11]:
Numeric=7.6705248490e-01
Analytic=7.6652583595e-01
iterations=11000
Energy [12]:
Numeric=8.9265853896e-01
Analytic=8.9208243070e-01
iterations=11000
Estamos em L = 79.29
Energy [0]:
Numeric=5.3492489436e-03
Analytic=5.3410051559e-03
iterations=72000
Energy [1]:
Numeric=2.1386931453e-02
Analytic=2.1358649816e-02
iterations=52000
Energy [2]:
Numeric=4.8089890176e-02
Analytic=4.8036446670e-02
iterations=33000
Energy [3]:
Numeric=8.5427628968e-02
Analytic=8.5345597938e-02
iterations=29000
Energy [4]:
Numeric=1.3336322337e-01
Analytic=1.3324275933e-01
iterations=22000
Energy [5]:
Numeric=1.9183524496e-01
Analytic=1.9166629646e-01
iterations=20000
Energy [6]:
Numeric=2.6075797944e-01
Analytic=2.6053013467e-01
iterations=16000
Energy [7]:
Numeric=3.3999773210e-01
Analytic=3.3971341731e-01
iterations=15000
Energy [8]:
Numeric=4.2937605996e-01
Analytic=4.2904218235e-01
iterations=13000
Energy [9]:
Numeric=5.2863206523e-01
Analytic=5.2825413616e-01
iterations=13000
Energy [10]:
Numeric=6.3737943553e-01
Analytic=6.3692219911e-01
iterations=11000
Energy [11]:
Numeric=7.5477297785e-01
Analytic=7.5425403472e-01
iterations=11000
Energy [12]:
Numeric=8.7890454306e-01
Analytic=8.7835257568e-01
iterations=11000
Energy [13]:
Numeric=9.9979896431e-01
Analytic=9.9766321067e-01
iterations=24000
Estamos em L = 80.00
In [23]:
L,e=tuple(zip(*eigenvalues))
l_e = pd.DataFrame({'L':L, 'e':e})
l_e.to_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv')
In [5]:
numeric = pd.read_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv', index_col=0)
In [6]:
ax = numeric[(numeric['L'] <= 80) & (numeric['e'] <= 1.0)].plot.scatter(x='L',y='e')
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_numerico.png', bbox_inches='tight')
In [7]:
analitic = pd.read_csv('saidas/poco_quantico_autovalores_por_comprimento.csv', index_col=0)
ax = analitic[analitic['L'] <= 80].plot.scatter(x='L',y='e')
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_analitico.png', bbox_inches='tight')
In [8]:
numcut = pd.read_csv('saidas/poco_quantico_autovalores_numericos_por_comprimento.csv', index_col=0)
numcut = numcut[(numcut['L'] <= 80) & (numcut['e'] <= 1.0)]
anacut = pd.read_csv('saidas/poco_quantico_autovalores_por_comprimento.csv', index_col=0)
anacut = anacut[(anacut['L'] <= 80) & (anacut['e'] <= 1.0)]
In [9]:
numcut['type'] = ['num']*numcut.shape[0]
anacut['type'] = ['ana']*anacut.shape[0]
In [10]:
fig, ax = plt.subplots(1,1)
ax.scatter(anacut['L'], anacut['e'], marker='o',c='k', s=40)
ax.scatter(numcut['L'], numcut['e'], marker='o',c='w', s=20)
ax.set_xlabel(r'a (\AA)')
ax.set_ylabel(r'E (eV)')
ax.set_ylim([-0.05, 1.05])
ax.set_xlim([8, 82])
plt.savefig('figuras/poco_quantico_numerico_e_analitico.png', bbox_inches='tight')
Poço de 50 A¶
In [11]:
poco_50 = PocoQuantico(well_length=50.0, well_height=1.0, N=2048, dt=1e-18)
In [12]:
fix, ax = plt.subplots(1,1, figsize=(18,6))
ax.plot(poco_50.z_ang, poco_50.v_ev)
ax.set_xlabel(r'z (A)')
ax.set_ylabel(r'V (eV)')
plt.show()
In [13]:
poco_50.evolucao_imaginaria(precision=1e-4)
Energy [0]:
Numeric=1.2951868994e-02
Analytic=1.2937397611e-02
iterations=33000
Energy [1]:
Numeric=5.1752222957e-02
Analytic=5.1699753783e-02
iterations=24000
Energy [2]:
Numeric=1.1623530904e-01
Analytic=1.1612905356e-01
iterations=16000
Energy [3]:
Numeric=2.0610841151e-01
Analytic=2.0592968138e-01
iterations=14000
Energy [4]:
Numeric=3.2087362193e-01
Analytic=3.2060263426e-01
iterations=11000
Energy [5]:
Numeric=4.5967820296e-01
Analytic=4.5929848646e-01
iterations=10000
Energy [6]:
Numeric=6.2092866745e-01
Analytic=6.2043969392e-01
iterations=9000
Energy [7]:
Numeric=8.0102193683e-01
Analytic=8.0041283309e-01
iterations=9000
Energy [8]:
Numeric=9.8405183319e-01
Analytic=9.8285878862e-01
iterations=18000
Out[13]:
<core.poco_quadrado_finito.PocoQuantico at 0x7f0dbe5b1940>
In [14]:
fix, ax = plt.subplots(1,1)
vsize = np.ptp(poco_50.v_ev)
ax.plot(poco_50.z_ang, poco_50.v_ev, label=r'$V(z)$')
for i, es in enumerate(poco_50.eigenstates):
psi2 = np.abs(es)**2
psi2 = 0.02 * psi2 * vsize / np.max(psi2) + poco_50.eigenvalues[i]
ax.plot(poco_50.z_ang, psi2, label=r'$|\psi_{%d} (z)|^2$' % i)
ax.set_xlabel(r'$z$ (\AA)')
ax.set_ylabel(r'$E$ (eV)')
plt.legend(bbox_to_anchor=(0,1.02,1,0.2), loc="lower left",
mode="expand", borderaxespad=0, ncol=5)
plt.savefig('figuras/poco_quantico_numerico_autoestados.png', bbox_inches='tight')
